Download Shortest paths and geodesics

Shortest paths and geodesics
in metric spaces
Nicklas Persson
Umeå Universitet
Handledare: Linus Carlsson
Examensarbete: 30 högskolepoäng
HT 2011
SHORTEST PATHS AND GEODESICS
IN METRIC SPACES
Nicklas Persson
Nicklas Persson: Shortest paths and geodesics in metric spaces,
c
Master’s thesis in mathematics, November
2012
Institution:
Mathematics and mathematical statistics
Supervisor:
Linus Carlsson
Examiner:
Per-Anders Boo
Location:
Umeå, Sweden
Time frame:
November 2012
Abstract
This thesis is divided into three part, the rst part concerns metric spaces
and specically length spaces where the existence of shortest path between
points is the main focus. In the second part, an example of a length space,
the Riemannian geometry will be given. Here both a classical approach
to Riemannian geometry will be given together with specic results when
considered as a metric space. In the third part, the Finsler geometry will
be examined both with a classical approach and trying to deal with it as
a metric space.
Sammanfattning
Denna uppsats är indelade i tre delar. Den första behandlar metriska
rum med betoning på längdrum där existensen av kortaste vägar är huvudsyftet. Den andra delen ger ett exempel på ett längdrum, Riemanska
geometrin. Här kommer både en klassisk upplägg till den Riemanska geometrin att ges tillsamanns med resultat där den är betraktad som ett
metriskt rum. I den tredje delen betraktas Finslergeometrin utifrån både
i ett klassik upplägg och ett försök ges att behandla den som ett metriskt
rum.
vi
Acknowledgements
I would like to thank my supervisor Linus Carlsson for his many hours
spent helping and listening to my many questions and my examiner PerAnders Boo who has taken time reading through this thesis and pointing
out errors and commenting on how to improve it. Without their help the
result would not have been nearly as good. Also I wish to thank Alexander
Zdunek for his time spent reading this work in its early state and coming
with constructive criticism.
vii
Contents
1 Introduction
1.1
Basic notation
. . . . . . . . . . . . . . . . . . . . . . . . . . . .
2 Metric geometry
2.1
2.2
2.3
Paths . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Length spaces . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Shortest paths and geodesics . . . . . . . . . . . . . . . . . . . .
3 Riemannian geometry
3.1
3.2
3.3
3.4
Manifolds . . . . . . . .
Riemannian metrics . .
Riemannian connections
Geodesics . . . . . . . .
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1
1
3
3
5
13
25
25
44
47
55
4 Finsler geometry
59
Appendices
73
A Metric spaces
73
B Topological spaces and topology
79
4.1
4.2
4.3
Finsler manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . .
Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
Geodesics . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
B.1 Topological spaces . . . . . . . . . . . . . . . . . . . . . . . . . .
B.2 Compactness . . . . . . . . . . . . . . . . . . . . . . . . . . . . .
59
61
66
79
86
C Algebra
105
D Vector spaces
113
E Functions
133
F Euclidean geometry
137
G Notation
139
H Bibliography
143
C.1 Linear algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 109
D.1 Dierential forms . . . . . . . . . . . . . . . . . . . . . . . . . . . 120
viii
List of Figures
1
2
3
4
5
6
7
8
9
10
11
12
13
14
15
Example of a non-length space . . . .
Example of another non-length space .
A sequence fn for n = 1, 2, . . . , 6 . . .
A geodesic γ . . . . . . . . . . . . . .
An atlas . . . . . . . . . . . . . . . . .
A unit sphere . . . . . . . . . . . . . .
A tangent bundle . . . . . . . . . . . .
A vector eld . . . . . . . . . . . . . .
Domain of germs . . . . . . . . . . . .
Tangent bundle . . . . . . . . . . . . .
A continuous function . . . . . . . . .
A counting of the rationals . . . . . .
An element of a base . . . . . . . . . .
A non convex set . . . . . . . . . . . .
Domain and codomain of a function f
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1 INTRODUCTION
1
Introduction
In this thesis, an introduction to the theory of metric geometry will be given.
Metric geometry deals with the geometrical topics which has a relation to the
notion of distance. During the 20:th century a lot of the theory developed in
geometry concerned the analysis of manifolds but in later years a lot of the work
done in the dierential geometry has shown that a metrical approach can be
conducted to reach the earlier results.
The results in this thesis are divided in three chapters. The rst gives an introduction to the general theory of metric geometry with emphasis on length spaces,
the existence of shortest paths and geodesics. The following two chapters gives a
brief classical approach to Riemannian geometry and Finsler geometry together
with attempts at trying to deal with them as metric spaces and studying the
existence of shortest paths. Some material which does not directly relate to
the main results of this thesis are collected into seven appendices which can be
found directly after the three main chapters where the last Appendix G gives a
list of the notations used here. The thesis end with an index and a bibliography.
An experienced reader of the topics covered can skip the material in the appendices or fast skim trough the material but readers with less experience can
with benet start by reading the appendices recommended before each chapter
and whenever an unknown term is used, look it up where it can be found in the
appendices.
1.1 Basic notation
Here will be given some basic notation used in the thesis. Vectors will be denoted
by [1, . . . , n] and not to confuse vectors of the form [a, b] with closed intervals
between a and b, the non-conventional denotation I[a,b] will be used for such
intervals. The letter d will be used for dierent metric functions, γ corresponds
to dierent curves and l(γ) is the length of the curve γ .
An open ball will be denoted by Bd (x, r) where d corresponds to the metric, x
the center of the ball and r is the radius. Closed balls will in a similar manner
be denoted by Bd (x, r). By Rn , the set of n-tuples of the real numbers R is
intended and E n corresponds to the Euclidean space which is Rn together with
an Euclidean structure which is basically an Euclidean inner product, norm and
metric.
The letter M will denote a manifold, Tp M will denote the tangent space of M
at a point p and T M the tangent bundle of M . The function g will be used for
Riemannian metrics and the corresponding object in the Finsler geometry, the
Finsler structure is denoted by F . For notations not covered in this short guide,
the list of notations in Appendix G is recommended.
1
2 METRIC GEOMETRY
2
Metric geometry
In this chapter there will be given a short introduction to the theory of metric
spaces with emphasis on the theory of length spaces and geodesics. The goal
of this chapter is to prove the Hopf-Rinow-Cohn-Vossen theorem which gives
a couple of equivalent statements for which a length space has shortest paths
between every pair of points.
To read and understand this chapter some prerequisite knowledge of topology,
topological spaces and metric spaces are needed. For those who are not so familiar with these terms or need refreshing these topics Appendix A deals with
most of the knowledge needed for metric spaces and Appendix B contains the
theory for topological spaces and topology. The goal of the appendices in this
thesis is to make it understandable for a reader who has studied two semesters
of mathematics without having to use exterior material. For those interested
in more detailed studies in the subjects of this chapter and the aforementioned
appendices the book [BBI01] deals with metric geometry, where most of the
results in this chapter can be found and the books [Rud76], [Mor05] are useful
for metric spaces and topology. Where [Mor05] is a good starting point for rst
time students of these subjects and [Rud76] for the more advanced.
In this chapter the standard topology (see Denition B.1.6) will be assumed for
topological spaces.
2.1 Paths
To later dene length spaces and the notion of geodesics the concept of paths is
essential. Paths are continuous maps of intervals and important to observe not
the image of the map. Formally they are dened as following.
Denition 2.1.1 (Path).
f : I[a,b] → X .
A
path in a topological space X is a continuous map
The almost equivalent term to path, curve (see Denition B.2.7) which is more
general will also be used for paths in this chapter.
Example 2.1.1 (Path). Given the topological space R2 . The curve
γ : I[0,1] → R2 , given by:
(
x = t2 ,
γ(t) =
y = t3 ,
t ∈ I[0,1]
t ∈ I[0,1]
(2.1.1)
is a continuous map between the points [0, 0] and [1, 1]. So γ is a path between
[0, 0] and [1, 1].
Example 2.1.2 (Path). Given the topological space R2 . The curve
given by:
γ : I[0,1] → R2 ,
3
2.1 Paths
2 METRIC GEOMETRY

2

x = t ,


x = t,
γ(t) =

x = 2t + 23 ,



y = t 3 ,
t ∈ I[0, 31 ]
t ∈ I( 31 , 23 )
t ∈ I[ 32 ,1]
t ∈ I[0,1]
(2.1.2)
is not a continuous map because the map is discontinuous at t = 31 and therefore
γ is not a path.
The concept of equivalent paths is important to be able to speak of converging
paths later on. Heuristically two path should be equivalent if the collection of
points visited by the paths are the same and the points are visited in the same
order. Furthermore if one of the paths stops at one point for a time while the
other continues but otherwise are similar then they should be considered equal.
Combining these properties and formalizing them gives the below denition.
Denition 2.1.2 (Equivalent paths). Two paths γ1 and γ2 are said to be equivalent if they belong to the same equivalence class dened by the equivalence
relation:
γ1 : I[a,b] → X and γ2 : I[c,d] → X are equivalent whenever there exists a
nondecreasing and continuous map θ : I[a,b] → I[c,d] such that:
(2.1.3)
γ1 = γ2 ◦ θ
Remark 2.1.1. Paths of the same equivalence class are called parametrizations
or re-parametrizations of one another and each are of the same length.
Example 2.1.3 (Equivalent paths). The paths γ1 : I[0,1] → R2 , where
(
γ1 (t) =
x=t
y=t
(2.1.4)
t
2
t
2
(2.1.5)
and γ2 : I[0,2] → R2 , where
(
x=
y=
γ1 (t) =
are equivalent.
Let θ : I[0,1] → θ : I[0,2]
γ1 = γ2 ◦ θ
be dened by: θ(t) = 2t. This continuous map satises:
Example 2.1.4 (Equivalent paths). The path γ1 : I[
(
x = r cos(t)
γ1 (t) =
y = r sin(t)
4
π
3π
4 ,− 4 ]
→ R2 ,
where
(2.1.6)
2.2 Length spaces
2 METRIC GEOMETRY
is a parametrization of the quarter-circle
x2 + y 2 = r 2 .
(2.1.7)
Another equivalent path of this quarter-circle is γ2 : I[−1,1] → R2 , where:
(
tr
x = √1+t
2
γ2 (t) =
(2.1.8)
r
y = √1+t2
That this is another parametrization can be seen by implicit derivation on
x2 + y 2 = r2 which gives:
d
2
2
dx (x + y
dy
x
dx = − y
dy
= r2 ) = 2x + 2y dx
= 0 ⇐⇒
dy
Setting −t = dx
gives that: x = yt, y = xt , inserting these in x2 + y2 = r2 , solve
for x and y gives (2.1.8).
2.2 Length spaces
The intrinsic metric or length metric is a metric possible to dene on every
metric space. For this metric the distance between two points is the length of
the "shortest path" between these these points. The term shortest path will be
dened later and is in fact crucial for the understanding of geodesics.
Denition 2.2.1 (Intrinsic metric). Given a topological space X with a metric.
The intrinsic metric (length metric) dI (x, y) for x, y ∈ X is given by:
dI (x, y) = inf l(γ)
γ∈P
(2.2.1)
Here P is the set of paths from x to y and if there is no path in X of nite
length between x and y , then set dI (x, y) = ∞ and l(γ) is the length of γ .
Denition 2.2.2 (Shortest path). Given a curve γ : I[a,b] → X . If γ is a path
and l(γ̃) ≥ l(γ) for every path γ̃ ∈ X such that the end points of γ̃ are γ(a) and
γ(b). Then γ is a shortest path.
Proposition 2.2.1. The shortest path between two points x, y ∈ E n is given by
a straight line between x and y.
Proof. Assume there exists a shorter path γ̃ between x, y than the straight line
γ . From the denition of the length of a curve B.2.8 we get the following equation:
l(γ̃) =
n−1
X
sup
a=t1 <t2 ...<tn =b i=1
5
d(γ̃(ti ), γ̃(ti+1 ))
(2.2.2)
2.2 Length spaces
2 METRIC GEOMETRY
But the partition t1 = a, t2 = b is the straight line between x, y and since
l(γ̃) is the supremum of the possible partitions: l(γ̃) ≥ d(x, y). This gives a
contradiction and therefore γ is a shortest path.
Example 2.2.1 (Intrinsic metric). Given the topological space E 2 . In E 2 , the
shortest paths are straight lines due to Proposition 2.2.1. The intrinsic √
metric
dI between the two points x = [0, 0] and y = [3, 3] is given by dI (x, y) = 18.
When the intrinsic metric is dened, the concept of length spaces is straightforward when the intrinsic and "usual" metric on the given space coincide. Further
on in this thesis almost every space considered will be length spaces and when
dealing with the question if shortest paths exist between dierent points in a
space, the whole theory in this chapter build on it being a length space.
Denition 2.2.3
(Length space). Given a topological space X with a metric
d. The space X is a length space if for every x, y ∈ X :
d(x, y) = dI (x, y),
(2.2.3)
where dI (x, y) is the intrinsic metric.
Example 2.2.2 (Length space). The Euclidean space E n is a length space. As
seen in Proposition 2.2.1 the shortest path in dI (x, y) is a straight line between
x and y and in this case the Euclidean metric d(x, y) coincides with dI (x, y).
Example 2.2.3 (Length space). The sphere S1 ⊂ R2 with the Euclidean metric
is not a length space. This can be seen by observing that paths in S1 are parts
of the sphere while the Euclidean metric gives rise to "chordals" between points
which are not on the sphere and d(x, y) ≤ dI (x, y) if x 6= y(see Figure 1).
Example 2.2.4 (Length space). Given a connected subset X in E 2 . If X is
convex the set is clearly a length space but if X is not convex than it is not a
length space. Then there exist two points x, y ∈ X such at there do not exists a
straight line between them which is in X . For these points dI (x, y) 6= d(x, y) as
seen for example in Figure 2.
From these examples the reader should now have a feeling of what length spaces
are and that the distance between points from the metric in length spaces can
informally be thought of as being measured inside the space.
A curve being natural is a property dependent of the parametrization of the
curve. Natural curves will be used in this chapter as a theoretical tool among
other things to prove the Arzela-Ascoli Theorem 2.3.1. What makes these type
of curves interesting beside simplifying certain computations are that for a given
curve there exists a parametrization which is natural as is shown in Proposition
2.2.2.
6
2.2 Length spaces
2 METRIC GEOMETRY
Figure 1: Example of a non-length space
Denition 2.2.4
speed) if:
(Natural curve). A curve γ : I[a,b] → X is
natural (unit
l γ(c), γ(d) = d − c for every c, d ∈ I[a,b] , where by l γ(c), γ(d) means the
length of the curve segment from γ(c) to γ(d).
Remark 2.2.1. The name unit speed parametrization for a natural parametrization comes from the fact that:
d
l(γ(a), γ(t)) = 1.
dt
(2.2.4)
A parametrization which satises:
(γ(a), γ(t)) = c(t − a)
is called a constant speed parametrization of speed c.
2
Example
( 2.2.5 (Natural curve). The curve γ : I[0,1] → R , where
γ(t) =
x=t
y=t
is not a natural curve. For example
7
(2.2.5)
2.2 Length spaces
2 METRIC GEOMETRY
Figure 2: Example of another non-length space
=
d
(γ(0), γ(t)) =
dtp
d
(t − 0)2 + (t
dt
− 0)2 =
√
2 6= 1,
for t > 0.
But
√ this parametrization is instead a constant speed parametrization with speed
2.
The notion of uniform convergence is a cornerstone in analysis and will be used
repeatedly later on. The dierence from pointwise convergence is informally
that converging uniformly has to do with how it converges over all of its domain
and for pointwise it is sucient that it converges at every point.
Formally pointwise convergence is stated as: Given a metric space (X, d) and
a subset Y . The sequence of functions fn : Y → X is said to be be pointwise
convergent on Y to the function f : Y → X if for every x ∈ Y :
there exists, for every > 0 an N such that for every n ≥ N ,
d(fn (x), f (x)) < .
(2.2.6)
This will further on be denoted as:
lim fn (x) = f (x).
n→∞
8
(2.2.7)
2.2 Length spaces
2 METRIC GEOMETRY
Denition 2.2.5
(Uniform convergence). Given a metric space (X, d) and a
set Y . A sequence of functions {fi }, where fi : Y → X is uniformly convergent
with limit f : Y → X if:
For every > 0, there exists an N ∈ N such that for every x ∈ Y and n ≥ N ,
(2.2.8)
d(fn (x), f (x)) < .
Example 2.2.6 (Uniform convergence). Given the set
metric. The sequence of functions {fn },
R
with the Euclidean
(2.2.9)
fn : I[0,1] → I[0,1]
where fn = xn is not uniformly convergent. This function instead satisfy the
weaker condition of point-wise convergence and converges to
(
f (x) =
0,
1,
if 0 ≤ x < 1
.
if x = 1
Assume that {fn } converges uniformly, choose =
an N ∈ N, such that:
|fn (x) − f (x)| <
1
2
(2.2.10)
. Then there should exist
1
2
(2.2.11)
for every x ∈ I[0,1] and all n ≥ N .
But choosing:
1
1>x>
1 N +1
2
and n = N + 1.
(2.2.12)
1
2
(2.2.13)
gives a contradiction due to:
|fN +1 (x) − f (x)| = |xN +1 | >
so {fn } is not uniformly convergent.
Informally it can be seen that fn does not converge uniformly by inspecting the
graphs (see Figure 3) of fn and noticing that a problem will arise due to the
discontinuity at x = 1 which will cause that for every < 1 there is an N such
that for x "close" to 1:
d(fn (x), f (x)) > (2.2.14)
The uniform convergence will now be used to dene uniform convergence of
curves which is crucial to prove the Arzela-Ascoli Theorem 2.3.1 later on.
9
2.2 Length spaces
2 METRIC GEOMETRY
Plot of fn (x)
1
n=1
n=2
n=3
n=4
n=5
n=6
0.9
0.8
0.7
fn (x)
0.6
0.5
0.4
0.3
0.2
0.1
0
0
0.1
0.2
0.3
0.4
0.5
1≤x≤1
0.6
0.7
0.8
0.9
1
Figure 3: A sequence fn for n = 1, 2, . . . , 6
Denition 2.2.6 (Uniform convergence of curves). A sequence of curves {γi }
converges uniformly to a curve γ if {γi } has a parameterization with domain
I[a,b] which uniformly converges to some parametrization of γ with domain I[a,b] .
Example 2.2.7 (Uniform convergence of curves). Given the curve
γ : I[0,1] → R2 , where
(
γ(t) =
x = t2
y=t
.
(2.2.15)
cos(t)
(2.2.16)
The sequence of curves γn : I[0,1] → R2 , where
(
x = t2 +
γn (t) =
y=t
1
n
converges uniformly to γ .
Example 2.2.8 (Uniform convergence of curves). Given the curve
where
(
γ : I[0,1] → R2 ,
γ(t) =
x = t2
y=t
.
(2.2.17)
The sequence of curves γn : I[0,2] → R2 , where
(
x=
γn (t) =
y=
converges uniformly to γ .
10
t2
2
t
2
+
1
n2
(2.2.18)
2.2 Length spaces
2 METRIC GEOMETRY
In this example the curves {γn } converges uniformly to a limit, which is a reparametrization of γ and this is sucient for {γn } to be uniformly convergent.
The aim of the following theory is to achieve results for when shortest paths
and the closely related term geodesics exists between dierent points in length
spaces. The two following propositions state the already mentioned result that
every curve has a re-parametrization which is natural and with a small modication of the proof that the curve has a re-parametrization with arbitrary speed.
Proposition 2.2.2. [BBI01] Given a rectiable (l(γ) < ∞) curve
γ : I[a,b] → X . It can be represented by:
γ = γ1 ◦ γ2 ,
(2.2.19)
where γ1 : I[0,l(γ)] → X is a natural curve and γ2 : I[a,b] → I[0,l(γ)] is nondecreasing and continuous map.
Proof. For constructing γ1 , let γ1 (τ ) be the point on γ such that the length of
the interval from its origin and to γ1 (τ ) is τ .
Now dene γ2 as:
γ2 (t) = l γ(a), γ(t)
(2.2.20)
for all t ∈ I[a,b] . This map is nondecreasing, continuous and its set of values is
I[0,l(γ)] .
The curve γ1 can be constructed by choosing a t ∈ I[a,b] for every τ ∈ I[0,l(γ)]
such that γ2 (t) = τ . Then dene γ1 (τ ) = γ(t) and thus γ1 : I[0,l(γ)] → X .
To show that γ1 is continuous, let τ1 = γ2 (t1 ) and τ2 = γ2 (t2 ). Now γ1 (τ1 ) and
γ1 (τ2 ) are end points of a path γ 0 which is the curve segment from γ1 (τ1 ) to
γ1 (τ2 ) of γ . The length of γ 0 is:
l(γ 0 ) = l(γ(t1 ), γ(t2 )) = γ2 (τ2 ) − γ2 (τ1 ) = τ2 − τ1 .
(2.2.21)
Utilizing that the distance between the endpoints in a length space is shorter
then the length of the path from Proposition 2.2.4 gives:
d(γ1 (τ1 ), γ1 (τ2 )) ≤ |τ1 − τ2 |
(2.2.22)
From (2.2.22) the map γ1 is Lipschitz continuous, this implies continuity by
Theorem A.0.9 and the below Remark A.0.9 and since γ 0 is a re-parametrization
of the path γ10 which is the curve segment from γ1 (τ1 ) to γ1 (τ2 ), the length of
γ10 is:
l(γ10 ) = l(γ(t1 ), γ(t2 )) = τ2 − τ1 .
(2.2.23)
This concludes that γ1 is a natural re-parametrization of γ .
11
2.2 Length spaces
2 METRIC GEOMETRY
Proposition 2.2.3. Given a rectiable curve
sented by
γ : I[a,b] → X .
It can be repre-
γ = γ1 ◦ γ2 ,
(2.2.24)
where γ1 : I[0,c·l(γ)] → X is a constant speed curve of speed c > 0 and γ2 :
I[a,b] → I[0,l(γ)] is nondecreasing and continuous map.
Proof. The proof is similar as the proof of Proposition 2.2.2. Choose
γ2 (t) = l(γ(a), γ(t))
(2.2.25)
as in Proposition 2.2.2 which will then be a nondecreasing and continuous map
and γ1 (τ ) = γ(t), where τ = cγ2 (t).
The following proposition gives two important properties for paths in length
spaces. These properties will be used repeatedly in dierent proofs in this chapter.
Proposition 2.2.4. [BBI01] Given a length space (X, d) and a path
γ : I[a,b] → X . Then the following statements are true:
a) l(γ) ≥ d(γ(a), γ(b)) (Triangle inequality)
b) Given a sequence of rectiable paths γi which converges pointwise to γ .
Then lim inf l(γi ) ≥ l(γ) (Semi-continuity of length)
i→∞
Remark 2.2.2. The operator lim inf called the limit inferior is dened for a
sequence xk as:
lim inf xk = sup inf xi .
(2.2.26)
k→∞
k∈N i≥k
This can be thought of as the smallest limit of a subsequence or if no such exists
±∞.
Proof.
The a) part of the proof is essential the same proof as for Proposition 2.2.1.
From the denition of length of curves B.2.8
l(γ) =
n−1
X
sup
d(γ(ti ), γ(ti+1 ))
(2.2.27)
a=t1 <t2 ...<tn =b i=1
But the partition a = t1 < b = t2 is part of the partitions of which the
supremum is taken so:
l(γ) ≥ d(γ(a), γ(b)).
(2.2.28)
12
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
b):
a): Choose > 0 and a partition P = {t0 , . . . , tN } of γ such that:
l(γ) −
N
X
d(γ(tj−1 ), γ(tj ) < (2.2.29)
j=1
Now consider:
Σ2 =
N
X
d(γi (tj−1 ), γi (tj ).
(2.2.30)
j=1
Choose i such that:
d(γi (tj ), γ(tj )) <
N
(2.2.31)
for every tj ∈ P and denote:
Σ1 =
N
X
d(γ(tj−1 ), γ(tj ).
(2.2.32)
j=1
By the triangle inequality:
d(γ(tj−1 ), γ(tj )) ≤ d(γ(tj−1 ), γi (tj−1 )) + d(γi (tj−1 ), γi (tj )) + d(γi (tj ), γ(tj ))
so we get:
|d(γ(tj−1 ), γ(tj )) − d(γi (tj−1 ), γi (tj ))| ≤
≤ d(γ(tj−1 ), γi (tj−1 )) + d(γ(tj ), γi (tj )) ≤
≤ 2/N
This gives:
l(γ) < Σ1 + ≤ Σ2 + + f rac(2N )N ≤ l(γi ) + 3
(2.2.33)
and since was chosen arbitrarily:
lim inf l(γi ) ≥ l(γ).
i→∞
2.3 Shortest paths and geodesics
Now the ground work is done to be able to show an important result from functional analysis which is here worked out for the special case of curves. The
concept of compactness will henceforth be very important and for those readers
which are not familiar with this it is recommended reading Appendix B.2 and
still a short look at this appendix is recommended for the more advanced reader
in order to see how the dierent kind of compactness are dened in this thesis.
13
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Theorem 2.3.1 (Arzela-Ascoli theorem for curves). [BBI01] Given a compact
metric space. Any sequence of curves which have uniformly bounded lengths has
an uniformly converging subsequence.
Remark 2.3.1. A sequence is bounded if every element in the sequence can be
bounded by the same ball.
Proof. From Proposition 2.2.3 there exists constant speed parametrizations of
γi on the interval I[0,1] . Because the length of γi is uniformly bounded and we
have constant speed, there exists a c < ∞ such that:
d(γi (t1 ), γi (t2 )) ≤ l(γ(t1 ), γ(t2 )) ≤ c|t1 − t2 |
(2.3.1)
for every i ∈ I and t1 , t2 ∈ I[a,b] .
Let S = {tj } be a countable dense subset of I[0,1] . From the Bolzano-Weierstrass
Theorem B.2.2, there is a subsequence γni of {γi } such that for each j ∈ N, the
sequence γni (tj ) converges. Now the goal is to show that γni converges. To
avoid double indices call γni = γi0 .
The sequence γi0 (t) is a Cauchy sequence for all t ∈ I[0,1] because: Given > 0,
choose tj ∈ S such that d(t − tj ) < and an N ∈ N such that:
d(γi0 (t − tj ), γk0 (tj )) < (2.3.2)
for every i, k > N . This gives that:
d(γi0 (t), γk0 (t)) ≤
≤ d(γi0 (t), γi0 (tj )) + d(γi0 (tj ), γk0 (tj )) + d(γk0 (tj ), γk0 (t)) ≤
≤ 3.
Because γi0 (t) is a Cauchy sequence we can dene
γ(t) = lim γj0 (tj ).
(2.3.3)
d(γ(t1 ), γ(t2 )) ≤ c|t1 − t2 |,
(2.3.4)
j→∞
Using (2.3.1) gives:
(2.3.4) gives that γ(t) is Lipschitz continuous and thus continuous due to Theorem A.0.4.
What is left to show now is that γi0 converges uniformly to γ : Given > 0,
choose N > 4c · and let M be such that:
d(γ(k/N ), γi0 (k/N )) <
, for all k = 0, 1, 2, . . . , N and i > M.
2
14
(2.3.5)
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
These choices are possible due to γi0 converging to γ pointwise.
Now using the results in (2.3.4) gives for all t ∈ I[0,1] and
k/N < t < (k + 1)/N :
(2.3.6)
d(γ(t), γi0 (k/N )) ≤
≤ d(γ(t), γ 0 (k/N )) + d(γ 0 (k/N ), γi0 (k/N )) ≤
≤ c|t − k/N | +
≤
4
+
2
+
4
2
+ c|t − k/N | ≤
≤
for every i > M . Thus the subsequence γi0 converges uniformly.
Here comes the rst result in this thesis which directly deals with shortest paths,
in this case that converging sequences of shortest paths are shortest paths and
this result is surprisingly easy to show.
Proposition 2.3.1. [BBI01] Given a sequence {γi } of shortest paths in a
length space (X, d) which converges to a path γ . Then γ is a shortest path.
Proof. Since the endpoints of γi converges to the endpoints a, b of γ and l(γi )
equals the distance between the endpoints, the length
l(γi ) → d(a, b).
(2.3.7)
Using the semi-continuity of length in Proposition 2.2.4 gives the following:
l(γ) ≤ lim l(γi ) = d(a, b)
i→∞
(2.3.8)
But d(a, b) is the distance of the shortest path between a and b so γ is a shortest
path.
The following proposition which states that in compact metric spaces there exist shortest paths between points which has rectiable curve between them is
an important result. The condition that the space needs to be compact is a
strong condition which can be weakened as will be seen later and the result of
Proposition 2.3.2 still holds.
Proposition 2.3.2. [BBI01] Given a compact metric space (X, d). Let
a, b ∈ X be points such that there exists a rectiable curve between a and b.
Then there exists a shortest path between a and b.
15
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Proof. Let linf be the inmum of lengths of rectiable curves between a and b.
There exists a sequence of such curves {γi } such that l(γi ) → linf . From ArzelaAscoli Theorem 2.3.1 there exists a converging subsequence {γni } of {γi } such
that {γni } will satisfy that:
l(γni ) → linf .
(2.3.9)
Let {γni } converge to a curve γ which will then have the end points a, b and
using 2.2.4 gives:
l(γ) ≤ lim l(γni ) = linf
i→∞
(2.3.10)
and thus we have that l(γ) = linf and γ is a shortest path.
Now to the problem of getting a stronger variant of Proposition 2.3.2. One way
to obtain such results are to dene boundedly compactness and later use these
spaces.
Denition 2.3.1 (Boundedly compact). Given a metric space (X, d). The
space X is boundedly compact if every closed and bounded subset of X is compact.
Example 2.3.1 (Boundedly compact). The space E n is a boundedly compact
space. This very fact is proven in the Heine-Borel theorem B.2.1.
Example 2.3.2 (Boundedly compact). The space Rn with the discrete metric:
(
0,
d(x, y) =
1,
if x = y
if x 6= y
(2.3.11)
and the standard topology is not boundedly compact. Choose a subset X ⊆ Rn
such that X contains innitely many points of (Rn , d). The open sets in this
topology are all subsets of Rn . Choose the points of X as a covering of X , then
clearly it does not exists a nite subcovering of X and thus X is not compact.
The subset X is bounded due to every set is bounded by a ball with radius r ≥ 1
and it is closed. This can be realized by choosing x ∈ X and the neighborhood
x. This neighborhood does not contain any other points in X and therefore it is
not an accumulation point. Thus X does not contain any accumulation points
and hence contain all of its accumulation points.
The following corollary is a stronger form of Proposition 2.3.2 where the condition of compact space is changed to the weaker condition of boundedly compact
space.
Corollary 2.3.1. [BBI01] Given a boundedly compact space (X, d). For every
x, y ∈ X such that there exists a rectiable curve between them there exists a
shortest path.
16
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Proof. Given points x, y ∈ X and let l be the length of the curve between them.
Considering the closed ball Bd (x, l), this is a compact set because Bd (x, l) is
closed, bounded and X is a boundedly compact space. Using Proposition 2.3.2
on the ball shows that there exists a shortest path between x and y .
Corollary 2.3.1 can be improved further by using local compactness dened
below instead of boundedly compactness. The goal now is to prove that completeness and local compactness implies boundedly compactness and boundedly
compact can be interchanged with complete and locally compact in Corollary
2.3.1.
Denition 2.3.2 (Locally compact). Given a topological space X . The space
X is locally compact if for every x ∈ X there exists a compact neighborhood of x.
Remark 2.3.2. Note that in the literature there are several dierent and nonequivalent denitions of local compactness. Among others, a common denition
is that every point instead of a compact neighborhood has a neighborhood whose
closure is compact (precompact). These denitions are not equivalent generally
but they are in Hausdor spaces and this is true for all other common denitions
of local compactness. The denition used in this thesis is implied by the second
denition given in this remark and this denition could instead have been used
in this thesis almost without any changes in the proofs.
Example 2.3.3 (Locally compact). The space E n is locally compact. This is
easily seen by choosing x ∈ E n and then choosing the compact neighborhood
Bd (x, r) of x.
Example 2.3.4 (Locally compact). The space
{[0, 0, 0]} ∪ {[x, y, z] : x > 0}
(2.3.12)
is not locally compact. For the point [0, 0, 0], there is no compact neighborhood.
Next to show that complete and locally compact implies boundedly compact we
generalize the notion of sequences with the concept of nets and use nets to show
the implication.
Denition 2.3.3 (Net). Given a metric space (X, d). A subset Y
an -net of X if for every x ∈ X , d(x, y) < for some y ∈ Y .
⊂ X is called
Example 2.3.5 (Net). Let the metric space (X, d) be the the closed ball X =
Bd (0, r + ) ⊂ E n with the euclidean metric. An -net to X is given by:
Y = Bd (0, r0 ),
where r < r0 < r + .
17
(2.3.13)
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Example 2.3.6 (Net). Given the metric space (R, d), where d is the Euclidean
metric. The subset Q ⊂ R is an -net of R for an arbitrary chosen > 0.
Before showing that that complete and local compactness implies bounded compactness the following lemma is needed.
Lemma 2.3.1. Given a closed set X in a complete metric space which is compact.
Then there exists a nite -net of X for every > 0.
Proof. Fix > 0 and x1 ∈ X . Now choose x2 ∈ X such that:
d(x1 , x2 ) ≥ .
(2.3.14)
Continue choosing x3 , x4 , . . . , xj+1 such that:
d(xi , xj+1 ) ≥ for every i = 3, 4, . . . , j.
(2.3.15)
Now every x1 , x2 , . . . , xj+1 are of distance or more from each other but this
construction can not continue innitely because using Theorem B.2.3 gives that
X is sequentially compact and thus every sequence in X has a convergent subsequence with limit in X . This gives a contradiction because every x1 , x2 , . . . , xj+1
are of distance or more from each other and thus for some j , x1 , x2 , . . . , xj
there is a pair xk , xl such that
d(xk , xl ) < .
(2.3.16)
Now choosing xj+1 ∈ X will give:
d(xj+1 , xi ) < for some i = 1, 2, . . . , j.
(2.3.17)
Continue choosing xj+2 , xj+3 , . . . , xj+h until:
d(xk , xl ) < for every pair xk , xl .
(2.3.18)
This can be done by choosing xj+i in the following way: Let
r = max {min {d(xi , xj ) : for j = 1, 2, . . . , j + 1 and j 6= i}} .
i
(2.3.19)
Then take the xi with the minimum index such that:
min {d(xi , xj ) : for j = 1, 2, . . . , j + 1 and j 6= i} = r
(2.3.20)
and denote as xk . Finally choose xj+i ∈ Bd (xk , r). Now the nite set {xi }i=1,2,...,j+h
is an net.
18
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Proposition 2.3.3. [BBI01] Given a complete locally compact length space
(X, d). Then every closed ball in X is compact.
Remark 2.3.3. Note that Proposition 2.3.3 implies that the complete locally
compact length space is boundedly compact because every closed and bounded
set Y ⊂ X can be contained in a closed, hence compact ball and using Lemma
B.2.2 gives that Y is compact.
Proof. Choose x ∈ X and note that if the closed ball Bd (x, r) is compact, so is
Bd (x, r0 ) for r0 < r
(2.3.21)
because Bd (x, r0 ) is closed, a subset of a compact set and hence compact from
Lemma B.2.2.
Let
n
o
R = sup r > 0 : Bd (x, r) is compact .
(2.3.22)
Because X is locally compact there exists such a R > 0 and assume that R < ∞.
For convenience denote Bd (x, R) as B .
Since B is a closed set in a complete space it is sucient to show that there
exists a nite -net for every > 0, for B to be compact from Lemma 2.3.1.
From the construction of B , < R can be assumed. Now consider the closed
ball:
Bd (x, R − /3)
(2.3.23)
denoted as B 0 . From the construction of R the ball is compact and hence by
Lemma 2.3.1 it contains a nite -net denoted as E . Choose y ∈ B , this gives:
d(y, w) ≤ /3 for some w ∈ B 0
(2.3.24)
because X is a length space. Furthermore
d(w, z) < /3 for some z ∈ E.
(2.3.25)
This gives that d(y, w) < and hence B is closed.
For every y ∈ B there exists a compact neighborhood Uy . Choose a nite
collection {Uy }y∈Y that covers B which is possible due to Theorem B.2.4. Now
the union
[
U=
Uy
(2.3.26)
y∈Y
is a compact neighborhood for all elements in B . Construct a set U 0 such that
B is an -net of U 0 and U 0 is contained in U . Because X is a length space the
ball:
U 0 = Bd (x, R + )
(2.3.27)
19
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
is such a set for some > 0. Then
(2.3.28)
Bd (x, R + )
is a closed subset of a compact set U and hence compact from Lemma B.2.2.
This contradicts the choice of R and hence R = ∞ which give that every closed
ball in X is compact.
Now when it is shown that complete and local compactness implies boundedly
it is nally possible to strengthen the statement in Corollary 2.3.1 with the following theorem.
Theorem 2.3.2. [BBI01] Given a complete locally compact length space (X, d).
Then for every x, y ∈ X such that d(x, y) < ∞, there exists a shortest path γ
from x to y.
Proof. From Proposition 2.3.3, (X, d) is a boundedly compact space and using
Corollary 2.3.1 gives that there exists a shortest path between x and y if
d(x, y) < ∞ (rectiable).
The above theorem is a very important result in this thesis and in fact the key
result in this thesis the Hopf-Rinow-Cohn-Vossen Theorem 2.3.3 is the equivalence between the condition in Theorem 2.3.2 and three other conditions.
An important generalization of shortest paths are geodesics. The dierence
between these are informally that geodesics are paths which only locally are
shortest paths. The term historically comes from the science of geodesy which
examines and measures distance, size and shape on the earth. There the term
stood for shortest paths, i.e. great circles on the earth but was later generalized
to dierent geometries where it comes to be the "straight lines" in that geometry. For example in the general relativity the "straight lines" in the curved
space time are the paths of which particles move inuenced only of gravitation
and thus these paths are the geodesics in that geometry.
Denition 2.3.4 (Geodesic). Given a length space (X, d). A curve γ : I → X
is a geodesic if for every x ∈ I, there exists an interval I[a,b] ⊆ I such that I[a,b]
contains a neighborhood of x and with the property that the restriction of γ on
I[a,b] , γ|I[a,b] is a shortest path from γ(a) to γ(b).
Example 2.3.7 (Geodesic). Given the length space
2-dimensional unit sphere in R3 . Let the curve
γ : I → S2
(S2 , dI ),
where S2 is the
(2.3.29)
be a line segment of a great circle (circle on S2 with radius 1). Then γ is a
geodesic. This can easily be seen by choosing x ∈ S2 and then let the interval
I[a,b] be such that:
d(γ(a), γ(b)) ≤ π.
20
(2.3.30)
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Then γ|I[a,b] is a shortest path from γ(a) to γ(b) (see Figure 4).
Figure 4: A geodesic γ
Remark 2.3.4. From Example 2.3.7 it is clearly visible that it is not true that
every geodesic is a shortest path. If γ is chosen such that there exists an interval
I[a,b] in the above example with
d(γ(a), γ(b)) > π.
(2.3.31)
Then γ|I[a,b] is not a shortest path.
Earlier when the concept of shortest paths was dened, it was restricted to
curves of the form γ : I[a,b] → X . To generalize the notion for shortest paths to
be dened on non-closed intervals. The curve γ : I → X is a shortest path or a
minimal geodesic if for every interval:
I[a,b] ⊂ I,
the restriction γ|I[a,b] is a shortest path, i.e. satisfying Denition 2.2.2.
21
(2.3.32)
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Left to show now is only the main theorem of this chapter and also of the entire
thesis, the Hopf-Rinow-Cohn-Vossen theorem.
Theorem 2.3.3 (Hopf-Rinow-Cohn-Vossen theorem). [BBI01] Given a locally
compact length space (X, d). Then the four following statements are equivalent:
a) X is a complete space
b) X is a boundedly compact space
c) Every naturally parametrized geodesic γ : I[0,a) → X is extendable to a continuous path
γ 0 : I[0,a] → X .
d) There exists a point x ∈ X such that every shortest path γ : I[0,a) → X ,
where γ(0) = x is extendable to a continuous path γ 0 : I[0,a] → X .
Remark 2.3.5. Using Theorem 2.3.2 gives that all these four conditions implies
that for every x, y ∈ X such that d(x, y) < ∞ there is a shortest path between
x and y if X is a locally compact length space.
Proof. We will show this theorem in the following order: b) =⇒ a), a) =⇒
c), c) =⇒ d) and d) =⇒ b).
b) =⇒ a): Assume X is a boundedly compact space.
Because X is boundedly compact, every closed ball in X is compact. Now
choose a Cauchy-sequence {xk } with limit:
lim xk = a.
k→∞
(2.3.33)
For some N ∈ N, the closure of the set {xk }k>N , denoted by K is contained
in a closed and hence compact ball Bd (xN , r). Since K is closed it contain its
accumulation points and since a is an accumulation point to K ,
a ∈ Bd (xN , r)
(2.3.34)
and hence a ∈ X .
a) =⇒ c): Let γ : I[0,a) → X be a naturally parametrized geodesic. Then:
l(γ(c), γ(d) = d − c
(2.3.35)
Choose a converging sequence:
{ai } ∈ I[0,a) such that 0 ≤ ai < a
and converging to a. Let
γi0 : I[0,ai ] → X
22
(2.3.36)
(2.3.37)
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
be such that γk0 is the same curve as the restriction γ|I[0,ak ] . Due to (2.3.35), γi0
is a Cauchy sequence and thus it has a limit γ 0 : I[0,a] → X which is continuous.
c) =⇒ d): Assume that every geodesic γ : I[0,a) → X is extendable to a continuous path γ 0 : I[0,a] → X .
If γ : I[0,a) → X is a shortest path, then
γ(0) = x for some x ∈ X.
(2.3.38)
Because every shortest path is a geodesic, the curve γ is extendable to a curve
γ 0 : I[0,a] → X .
d) =⇒ b): Assume there exists a point x ∈ X such that every shortest path
γ : I[0,a) → X , where γ(0) = x is extendable to a continuous path γ 0 : I[0,a] → X .
The proof of this follows the general scheme of the proof of proposition 2.3.3.
Because X is locally compact, a suciently small closed ball Bd (x, r) is compact.
Let:
n
o
R = sup r : Bd (x, r) is compact
(2.3.39)
and assume R < ∞.
First prove that the open ball Bd (x, R) is precompact (its closure is compact)
which will give that Bd (x, R) is compact. From Theorem B.2.3 it is sucient
to show that every sequence {xi } contains a converging subsequence (note that
the limit does not necessary belong to the ball). Let ri = d(x, xi ) and assume:
lim ri = R
i→∞
(2.3.40)
which is possible because otherwise {xi } will be contained in a smaller compact
ball and that ball will have a converging subsequence.
Let γi : I[0,ri ] → X be a shortest path from x to xi . To see that such a path
exists see the proof of Corollary 2.3.1. Now choose a subsequence of {γi } such
that the restriction to the interval I[0,r1 ] is converging. Continue choosing from
this subsequence such that the restriction to I[0,r2 ] is converging and so on.
Then choosing the nth element of the nth subsequence (Cantor diagonalization
process) produces a sequence of paths {γni } such that for every t ∈ I[0,R) the
sequence {γni (t)} converges in X and the endpoints of {γni } is a subsequence
{xni } of {xi }.
Letting:
γ(t) = lim γni (t)
(2.3.41)
gives that γ is a shortest path from Proposition 2.3.1. From the assumption
there is an extended continuous path γ 0 : I[0,a] → X and the endpoints of the
converging curves {γni } converges to γ 0 (R). This gives that the subsequence
{xni } of {xi } converges to γ 0 (R) and hence Bd (x, R) is compact.
23
2.3 Shortest paths and geodesics
2 METRIC GEOMETRY
Now show that a ball:
Bd (x, R + )
(2.3.42)
for some is compact. Using that X is locally compact gives that there is a
compact ball Bd (y, r(y)) around y for every
y ∈ Bd (x, R). Choose a nite covering of Bd (x, R) by the balls:
n
o
Bd (y, r(y)) .
(2.3.43)
Then the union of the closed balls U is a compact set and there is a ball:
Bd (x, R + )
(2.3.44)
Bd (x, R) ⊂ Bd (x, R + ) ⊂ U.
(2.3.45)
for some such that:
Because Bd (x, R + ) is a closed subset of a compact set the Lemma B.2.2 gives
that Bd (x, R + ) is compact which contradicts the choice of R, hence R = ∞
and X is boundedly compact.
In the two following chapters Riemannian geometry and Finsler geometry the
theory from this chapter will be applied. The Riemannian geometry with the
right metric will be a length space and thus much of the theory from this chapter
will be applicable. In the Finsler geometry a problem arises because "choosing"
the right "metric" will not give a length space because the chosen function is
not a metric but a semi metric. This creates some problems because the HopfRinow-Cohn-Vossen theorem is not readily applicable but still it is possible to
obtain a result corresponding to the Hopf-Rinow-Cohn-Vossen theorem in the
Finsler geometry.
24
3 RIEMANNIAN GEOMETRY
3
Riemannian geometry
In 1854 Bernard Riemann held a famous lecture "Über die Hypothesen welche
der Geometrie zu Grunde liegen " where he lay the foundation of Riemannian
geometry by among other things generalizing the dierential geometry in R3
and lay the foundations of the theory of manifolds. Already during the rst
part of the 19th century a special case of a Riemannian geometry, the hyperbolic geometry, was studied independently by the mathematicians Lobachevsky,
Gauss and Bolyai.
The Riemannian geometry is a geometry characterized by a mathematical object called a manifold which locally resembles a subset of the n-dimensional
Euclidean space and a Riemannian metric which denes concepts like angles,
volumes, length of curves, etc. Together these object constitutes a Riemannian
manifold which is the object of focus in Riemannian geometry.
A real world example of what a Riemannian geometry is and why they are useful is the length of the path a hiker traverses in a mountainous region. Using
the Euclidean geometry the path traveled is "independent" of if the path goes
through rough terrain and thus is very tiresome or not. Instead if this would be
a Riemannian geometry the Riemannian metric would govern the length of the
paths by "penalizing" paths with rough terrain by adjusting the length path
by not only the length in the Euclidean sense but also how tiresome the path
is. Two possible interpretations of the length of these paths are either the time
traveled or the energy used by the hiker.
The content of this chapter is a classical approach of Riemannian geometry by
introducing the foundations of manifold theory, dening Riemannian manifolds
and the Riemannian connection in order to derive the Christoel symbols and
the geodesic equation. Then there is a section when the Riemannian geometry
is handled as a metric space where a metric is dened such that a length space
arises and the Hopf-Rinow-Cohn-Vossen theorem is applicable. For a more thorough exposure of Riemannian geometry and the analysis of manifolds the books
[GHL04] and [Küh02] are two good starting points where most of the theory in
this chapter can be found.
3.1 Manifolds
The theory of manifolds has been developed during the later half of the 19th
century and as already stated Riemann was one of the early pioneers in this
subject. One informal example of manifolds which is important both for understanding of where the notion originated from and how they work is the surface
of a sphere. In R3 this object is a 2-dimensional abstract manifold which will
be seen later. Why it is a 2-dimensional abstract manifold is because by using
subsets of R2 , called charts, it is possible to represent the surface of the sphere.
Think of these charts as an atlas covering the surface of the world and from this
analogy a collection of charts is named an atlas.
Why the theory of manifolds is so useful is mainly because the theory for Rn is
already well-developed. Complicated objects which are hard to study directly
then behaves mostly like a subset of Rn when treated as a manifold and for
25
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
example the results of calculus are mostly applicable.
In the rst section of this chapter the foundations of the theory of manifolds will
be dened. Abstract manifolds (dierentiable manifolds) will be used to dene
tangent spaces, tangents, the tangent bundle and then the tangent bundle is
used to dene vector elds. For those acquainted with dierential geometry
most of these terms will be familiar and only generalized to abstract manifolds.
Before dening abstract manifold, submanifolds of Rn will be dened which will
later be used to to dene submanifolds of abstract manifolds.
Denition 3.1.1 (Dierential map).
f=
f1
Given a function:
... f m : Rn → Rm
(3.1.1)
where f is C k (k times continuously dierentiable). Then the
of f at p is given by:


Dfp = 
∂f 1
∂x1 (p)
..
.
m
∂f
∂x1
...
(p) ...
∂f 1
∂xn (p)
..
.
m
∂f
∂xn
dierential map

(3.1.2)


(p)
Example 3.1.1 (Dierential map). The function f : R2 → R, given by
f (x, y) = x2 y , has the dierential map at p = [p1 , p2 ] given by:
(3.1.3)
Dfp = [2p1 p2 , p21 ].
The dierential for this function is given by:
(3.1.4)
dfp = 2p1 p2 dp1 + p21 dp2 .
Denition 3.1.2
(Submersion). A function f : M → N is a
dierential map is surjective for all p ∈ M .
Example 3.1.2 (Submersion). The function f
f (x, y) = x + y
: R 2 → R,
submersion if its
given by:
(3.1.5)
is a submersion. The dierential map:
Dfp : R2 → [1, 1]
(3.1.6)
at p = [p1 , p2 ] is given by: Dfp = [1, 1]. To show that Dfp is surjective, take
the only element [1, 1] in the codomain and show there exists an element v ∈ R2
which is mapped on [1, 1], but this is satised for every element v ∈ R2 and thus
Dfp is a submersion.
26
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Example 3.1.3 (Submersion). The function f
not a submersion. The dierential map:
: R → R,
given by f (x) = x3 is
(3.1.7)
Dfp : R → R
at p is given by: Dfp = [3p2 ]. For the element −1 ∈ R there is no x in the
domain such that 3x2 = −1 and thus Dfp is not a submersion.
Denition 3.1.3 (Submanifolds of Euclidean spaces).
A set:
M ⊂ Rn+m
(3.1.8)
is an n-dimensional submanifold of class C k of Rn+m if, for any p ∈ M , there
exists a neighborhood (B.1.9 on page 83) U of p in Rn+m and a C k submersion
f : U → Rm where
U ∩ M = f −1 (0).
(3.1.9)
Example 3.1.4 (Submanifold). Consider the object:
M = x = [x1 , x2 , x3 ] ∈ R3 : f (x) = 4x21 + 2x22 + x23 − 4 = 0
(3.1.10)
This is a 2-dimensional submanifold of class C ∞ of R3 . The dierential is given
by:
dfx = 8x1 dx1 + 4x2 dx2 + 2x3 dx3
(3.1.11)
The function f dened in (3.1.10) is a submersion around every point in M
due to the dierential map spanning R on M . Choose p ∈ M and let U be the
neighborhood
R3 \ Bd (0, ),
(3.1.12)
where Bd (0, ) is a small ball such that:
Then
Bd (0, ) ∩ M = ∅.
(3.1.13)
U ∩ M = f −1 (0).
(3.1.14)
Example 3.1.5 (Submanifold). Consider the object:
M = x = [x1 , x2 , x3 ] ∈ R3 : f (x) = 4x21 + 2x22 + x23 = 0
(3.1.15)
This object is not a 2-dimensional submanifold of class C ∞ of R3 . In fact
M = {0} so this should not be surprising. Furthermore the function f dened
in (3.1.15) is not a submersion around every point in M due to the dierential
map not spanning R at 0 ∈ R3 .
27
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Before dening abstract manifolds, two properties for functions are needed,
homeomorphism and dieomorphism. Homeomorphic functions are the counterpart to isomorphisms for topological spaces and thus such mappings preserves
the topological properties and the two spaces are essentially equivalent from a
topological point of view.
Denition 3.1.4 (Homeomorphism). A function f : X → Y is a homeomorphism between two topological spaces if the following is satised:
a) f is a bijection
b) f is continuous
c) f −1 is continuous
Example 3.1.6 (Homeomorphism). Considering the function:
f : (R, σ) → (R, σ),
(3.1.16)
where σ is the discrete topology (see denition B.1.4) and f is given by:
f (x) = 5x + 3.
(3.1.17)
f is obviously a bijection and to show that f is continuous, choose an arbitrary
V ∈ σ . For f to be continuous, f −1 (V ) has to be an open set in σ but this is
trivial due to σ consisting of every subset of R. The inverse is
f −1 (x) =
1
3
x−
5
5
(3.1.18)
which is a continuous function using the same logic as for f , so f is a homeomorphism.
Example 3.1.7 (Homeomorphism). Considering the function:
f : (X, σ1 ) → (Y, σ2 ),
(3.1.19)
where σ1 , σ2 are the discrete topologies of X , Y respectively,
X = {−1, 0, 1} , Y = {0, 1, 2}
(3.1.20)
and f is given by f (x) = x + 1. f is obviously a bijection and f is continuous.
The inverse is f −1 (x) = x − 1. This function is continuous so f is a homeomorphism.
Example 3.1.8 (Homeomorphism). Considering the function:
f : (X, σ1 ) → (Y, σ2 ),
28
(3.1.21)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
where σ1 , σ2 are the discrete topologies of X , Y respectively,
X = {−1, 0, 1} , Y = {−1, 0, 1, 2}
(3.1.22)
and f is given by f (x) = x + 1. Here f is not a bijection which was show in
Example E.0.18. Choosing −1 ∈ σ2 , now f −1 (−1) = −2 is not an open set in
σ1 so f is not continuous. The element −1 ∈ Y was not mapped on by any
element in X so therefore the inverse to f does not exists.
A dieomorphism is closely related to homeomorphisms. Instead of only demanding f and f −1 being continuous, they need to be continuously dierentiable and thus f also preserves the topological properties. The continuous
dierentiability property will later give the dierentiable structure needed in
the abstract manifold.
Denition 3.1.5 (Dieomorphism). Given two manifolds X and Y . A function
f : X → Y is a C k dieomorphism if:
a) f is a bijective function (Bijection)
b) f is k -times continuously dierentiable (Dierentiable)
c) f −1 is k -times continuously dierentiable (Dierentiable)
Example 3.1.9
given by:
(Dieomorphism).
Given the function f : R → R, where f is
f −1 (x) =
3
1
x− .
5
5
(3.1.23)
This function is clearly bijective and f, f −1 are smooth functions so f is a C ∞
dieomorphism.
The important step when creating an abstract manifold of a set is constructing
an appropriate atlas. An atlas consists of an open covering of the set, together
with a collection of functions which connect every element of the covering with
a subset of Rn and each function is a homeomorphism which preserve the topological properties in the original set to the new collection of subsets of Rn .
Furthermore some constraints in the choice of the homeomorphisms are needed
in order to get the dierentiability of a manifold. Formalizing this gives the
below denition.
Denition 3.1.6 (Atlas).
Given a Hausdor space X . A C k
atlas is given by:
a) An open covering, {Ui }i∈I of X , that is:
X ⊆ ∪i∈I Ui .
29
(3.1.24)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
b) A family of homeomorphisms fi : Ui → Ωi (where Ωi ⊂ Rn ), such that:
for all index i, j ∈ I ,
fj ◦ fi−1 |fi (Ui ∩Uj )
(3.1.25)
is a C k dieomorphism from fi (Ui ∩ Uj ) onto fj (Ui ∩ Uj ) (see Figure 5).
Figure 5: An atlas
Example 3.1.10 (Atlas). Considering the Hausdor space R with the standard
topology. The covering U1 = R together with the function f1 : U1 → R, given by
f1 (x) = x is a C ∞ atlas. This can be seen by noticing that in fact U1 is trivially
a covering of R, f1 is a homeomorphism with inverse
f1−1 (x) = x
(3.1.26)
satisfying b) in Denition 3.1.6. Here,
f1 ◦ f1−1 = f (x) = x
which is a C ∞ dieomorphism from f1 (U1 ) onto f1 (U1 ).
30
(3.1.27)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Example 3.1.11 (Atlas). Considering the Hausdor space R3 with the standard
topology and the Euclidean metric F.0.12. The unit sphere S2 is a 2-dimensional
object satisfying:
S2 = [x, y, z] ∈ R3 |x2 + y 2 + z 2 = 1
(3.1.28)
Constructing an atlas for S2 is possible with six charts. Considering the coverings Ui :
U1
U2
U3
U4
U5
U6
= [x, y, z] ∈ R3 |x2 + y 2 + z 2
= [x, y, z] ∈ R3 |x2 + y 2 + z 2
= [x, y, z] ∈ R3 |x2 + y 2 + z 2
= [x, y, z] ∈ R3 |x2 + y 2 + z 2
= [x, y, z] ∈ R3 |x2 + y 2 + z 2
= [x, y, z] ∈ R3 |x2 + y 2 + z 2
=1
=1
=1
=1
=1
=1
and z > 0
and z < 0 and x > 0
and x < 0
and y > 0
and y < 0
and the homeomorphism fi : Ui → D2 :
f1 (x, y, z) = [x, y]
f2 (x, y, z) = [x, y]
f3 (x, y, z) = [x, z]
f4 (x, y, z) = [x, z]
f5 (x, y, z) = [y, z]
f6 (x, y, z) = [y, z]
which maps the half-spheres on the unit disc D2 .
The open covering
6
[
Ui
(3.1.29)
i=1
covers S2 where for example U1 is the upper semi-sphere in Figure 6 and the
functions corresponding to the open covering are also homeomorphisms. For fi
to be a homeomorphism it needs to be a bijection. Here every element in Ui
maps exactly one element of D2 , every element in D2 was mapped on by exactly
one element in Ui so fi is a bijection which yields that f −1 exists. The functions
f and f −1 are continuous so f is a homeomorphism.
Here,
fj ◦ fi−1 |fi (Ui ∩Uj )
(3.1.30)
is clearly well-dened and goes from fi (Ui ∩ Uj ) to fj (Ui ∩ Uj ). For example let
i = 1 and j = 3, U1 ∩ U3 is the quarter-sphere dened by x ≥ 0 and z ≥ 0 (see
Figure 6). Then
f3 ◦ f1−1 |f1 (U1 ∩U3 )
(3.1.31)
has the domain f1 (U1 ∩ U3 ) and
f1−1 (f1 (U1 ∩ U3 )) = U1 ∩ U3 .
31
(3.1.32)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
1
0.5
0
−0.5
−1
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
Figure 6: A unit sphere
Because of this f3 (U1 ∩ U3 ) is the corresponding codomain.
Now checking this for all pairs of i, j following the same scheme. For this to be
an atlas
fj ◦ fi−1 |fi (Ui ∩Uj )
(3.1.33)
need to be a dieomorphism. Due to fi being a homeomorphism, fj ◦fi−1 |fi (Ui ∩Uj )
needs to be a bijection. Furthermore fj ◦fi−1 |fi (Ui ∩Uj ) and its inverse are smooth
functions so the pair {Ui } and {fi } is a C ∞ atlas to S2 .
Example 3.1.12 (Atlas). Given the unit circle S1 . By a similar reasoning as
for S2 in Example 3.1.11, an atlas for S1 can be created by:
U1
U2
U3
U4
= [x, y] ∈ R2 |x2 + y 2
= [x, y] ∈ R2 |x2 + y 2
= [x, y] ∈ R2 |x2 + y 2
= [x, y] ∈ R2 |x2 + y 2
=1
=1
=1
=1
and x > 0
and x < 0
and y > 0
and y < 0
and the homeomorphism fi : Ui → I[0,1] where I[0,1] is the closed interval between 0,1:
f1 (x, y) = y
f2 (x, y) = y
f3 (x, y) = x
f4 (x, y) = x
Using the same logic as for S2 the pair {Ui } and {fi } are a C ∞ atlas to S1 .
32
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Denition 3.1.7 (Maximal atlas). Given a Hausdor space X . Two C k atlases
of X are equivalent if the union of the atlases is a C k atlas of X . A maximal
atlas of X is an equivalence class of equivalent atlases of X .
Remark 3.1.1. The equivalence class which denes a maximal atlas is also known
as a dierentiable structure.
When a maximal atlas is constructed to a set the resulting pair is an abstract
manifold and the denition is formalized below.
Denition 3.1.8 (Abstract manifold).
An n-dimensional abstract C k
is a collection of a Hausdor space M and a maximal C k atlas of M .
manifold
Remark 3.1.2. When referring to an abstract manifold, one element of the equivalence class will be given and letting that element represent the equivalence class.
Example 3.1.13 (Abstract manifold). The set R together with the C ∞ atlas
in Example 3.1.10, (U1 , f1 ) is a 1-dimensional abstract C ∞ manifold.
Example 3.1.14 (Abstract manifold). The set S2 together with the C ∞ atlas
in Example 3.1.11 is a 2-dimensional abstract C ∞ manifold.
As mentioned in the introduction to this section, the atlas consists of a collection of charts. Each chart can be thought of as a page in a "paper-version" of
an atlas. Such a construction is given by an open set from the covering with its
corresponding function which together forms a chart.
Denition 3.1.9
(Chart). For a given index i ∈ I , a pair (Ui , fi ) , where
Ui ∈ M and fi satisfying Denition 3.1.6 are a chart of M .
Example 3.1.15 (Chart). The pair (U1 , f1 ) in Example 3.1.10 is a chart of R.
Example 3.1.16 (Chart). Every (Ui , fi ) in Example 3.1.11 is a chart for S2 .
Early on in this chapter submanifolds of Rn was dened in Denition 3.1.3.
Using these object a submanifold to an abstract manifold can be created by
choosing a subset of the manifold such that there exists, for any point in the
subset, a chart (f, U ) covering that point and such that f (U ) ∩ Rn is submanifold of an Euclidean space. This is formalized below.
Denition 3.1.10
(Abstract submanifold). Given a smooth n-dimensional
manifold M . A subset N ⊂ M is an m-dimensional abstract submanifold of
M if there exists for any p ∈ N a chart (U, f ) of M around p so f (U ∩ N ) is an
m-dimensional submanifold of f (U ) ∩ Rn .
33
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Example 3.1.17 (Abstract submanifold). Consider the upper half-sphere of
S2 given by U1 in Example 3.1.11. Using that Example, there is always one or
more charts which includes a p ∈ U1 . Choose one such chart, {U, f }. Clearly
f (U ∩U1 ) have to be a submanifold of f (U )∩R2 if U1 is an abstract submanifold
of S . The set f (U ) ∩ R2 will be the 2-dimensional unit disc D2 .
From here on unless otherwise stated an abstract manifold M will be assumed
smooth and n-dimensional.
When abstract manifolds are dened, the dierential structure can be used to
create tangent vectors and tangent spaces to a curve. A problem arises because
dening the tangent at a point γ(p) ∈ M as γ 0 (p) is not directly possible because
dierentiating is not dened. This is not a problem for submanifolds of Rn , for
which the denition is straightforward.
Denition 3.1.11 (Tangent in submanifold).
Given an n-dimensional submanifold M of Rn+k , p ∈ M and x ∈ Rn+k . Then x is a tangent of M at p if there
is a curve γ on M such that γ(0) = p and γ 0 (0) = x
Example 3.1.18 (Tangent in submanifold). Given the manifold S2 in Example
3.1.11 given by equation (3.1.28). This sphere is also a submanifold which can
be seen by using the same method as was used in Example 3.1.4. Considering
the curve:


x = sin(t),
γ(t) = y = 0,


z = cos(t),
t ∈ I[−π,π]
t ∈ I[−π,π]
t ∈ I[−π,π]
(3.1.34)
This curve is on the submanifold S2 and γ(0) = [0, 0, 1] = p. Furthermore:
γ 0 (0) = [0, 0, 0] = x
(3.1.35)
so x is a tangent at p to the submanifold S2 .
The tangent spaces to manifolds are intuitively the "directional derivatives" at
a point and if the tangents are dened the tangent space is dened as followed.
Denition 3.1.12
(Tangent space in submanifold). The set of tangents (tangent vectors) to M at p is called the tangent space of M at p, denoted Tp M .
Example 3.1.19 (Tangent space in submanifold). Once again considering the
sphere S2 in Example 3.1.11. The tangent space at a point p, Tp S is given by:
{[a, b, 0] | ∀a, b ∈ R}
(3.1.36)
which is the plane z = 1.
This can be seen by realizing that every curve γ on S2 passing through p has a
tangent vector of the form [a, b, 0] because the derivative of z at p will always be 0.
34
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
In this thesis two equivalent denitions of tangents of abstract manifolds will be
given. The rst denition is given below, which essentially says that a tangent
is a tangent to a curve lying on the manifold.
Denition 3.1.13 (Tangent in abstract manifold). Given an abstract manifold M and a point p ∈ M . A tangent vector of M at p is an equivalence class
(C.0.15 on page 106) of curves γ : I0 → M , where I0 is an interval containing 0
and γ(0) = p. The equivalence relation (∼) is dened by:
γ0 ∼ γ1 ⇐⇒ Given chart (U, f ) around p, the following holds:
(f ◦ γ1 )0 (0) = (f ◦ γ0 )0 (0)
(3.1.37)
This denition is easy to understand but poses two problems. Firstly, the property of being a tangent vector to a manifold is not clearly independent of the
choice of chart. Secondly, it is not possible to chose a single principal representative for an equivalence class of a tangent vector.
Example 3.1.20 (Tangent in abstract manifold). Given the sphere, S2 in Example 3.1.11. In Example 3.1.18 a tangent in the submanifold S2 was found.
Other such curves γ are given by:


x = sin(t)
γ(t) = y = 0


z = cos(t)
, t ∈ I[a,b]
, t ∈ I[a,b]
, t ∈ I[a,b]
(3.1.38)
where −π ≤ a < 0 and 0 < b ≤ π. All these curves belong to the same equivalence class, but these are not the only members for which the equivalence relation
is fullled.
As for the tangent space of submanifolds, the tangent space of an abstract manifold at a point is simply the set of tangent vectors which in this case is a set of
equivalence classes.
Denition 3.1.14
(Tangent space in abstract manifold). The set of tangents
(tangent vectors) to M at p is called the tangent space of M at p, denoted Tp M .
Example 3.1.21 (Tangent space in abstract manifold). The tangent space of
S2 at p = [0, 0, 1] is the set of equivalence classes for which the relation is fullled at p. Some of these curves gives rise to equivalence class given in Example
3.1.20. Furthermore every pair a, b in Example 3.1.19 will generate a curve
which will belong to one of the equivalence classes and every equivalence class
has a curve from Example 3.1.19.
The next object of interest is the tangent bundle. As the name indicates, the
tangent bundle is informally the space of all tangent spaces for a manifold. This
object is important in the Riemann geometry and as will be seen later, many
35
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
100
50
0
−50
−100
1
1
0.5
0.5
0
0
−0.5
−0.5
−1
−1
Figure 7: A tangent bundle
objects in the Riemannian and Finsler geometry are stated in terms of tangent
bundles.
Denition 3.1.15 (Tangent bundle). Let the tangent bundle T M of M be the
disjoint union of the tangent spaces to M for all p ∈ M .
Remark 3.1.3. The meaning of the word disjoint union above is that no two
tangent spaces can have common vectors among them. For this to be possible
more dimensions are needed. For an n-dimensional manifold the dimension of
the tangent bundle is 2n.
Example 3.1.22 (Tangent bundle). Given the manifold S1 in Example 3.1.12.
The tangent bundle of this object can be visualized by the Cartesian product
S1 × R. Geometrically this could be interpreted by an innitely high cylinder
consisting of a circle centered at origo and radius 1 seen in Figure 7 if visualized
being of innite height.
Once the tangent bundle is dened vector elds comes naturally as the "directional derivatives", as seen from the dierential geometry. Formalizing this
comes naturally in the Riemannian geometry as:
Denition 3.1.16
(Vector eld). Given a manifold M . A vector
called a smooth section of the bundle T M , i.e. a smooth map X :
36
eld in M is
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
{X : M → T M | for any p ∈ M, X(p) ∈ Tp M }
(3.1.39)
We will denote the set of all vector elds on M by Γ(T M ) and X(p) by Xp
Example 3.1.23 (Vector eld). Given the sphere S1 in Example 3.1.12. The
smooth map f : S1 → T S1 given by:
f (x, y) = g(x, y)[y, −x],
(3.1.40)
where g is of the form g(x, y) : R2 → R (see Figure 8). Then f is a vector eld
in S1 since f (x, y) is orthogonal (the inner product is zero) with [x, y]:
h[x, y], g(x, y)[y, −x]i = x · g(x, y) · y − y · g(x, y) · x = 0
and hence
f (x, y) ∈ T[x,y] S1 .
(3.1.41)
(3.1.42)
Another disadvantage of the denition now being used for tangents is that it
becomes unnecessarily hard to prove that tangent spaces are vector spaces. Because of this another denition of tangent will be given which states that tangent
vectors are derivations.
To dene a derivation, the concept of germs of functions is needed. The germs
at a point can be thought of as the equivalence class of functions for a given
function class (C 1 , C 2 , continuous, etc.) for which every function is equal in a
small neighborhood of the point. More precisely stated as:
Denition 3.1.17 (Germs).
Given a topological space X . The germs at p ∈ X
for a given function class A (ex. continuous, C k , smooth etc.) dened in a
neighborhood of p is the equivalence class of functions f ∈ A satisfying the
equivalence relation (∼), given by:
There is f1 : U → R and f2 : V → R such that there exists a neighborhood
W ⊂ (U ∩ V ) of p,
(3.1.43)
such that f1 (x) = f2 (x) for every x ∈ W (see Figure 9). This will denoted:
f1 |W = f2 |W
henceforth.
37
(3.1.44)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Figure 8: A vector eld
38
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Figure 9: Domain of germs
39
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Notation 3.1.1 (Cpk (M )). The function space Cpk (M ) are the germs of functions of the class C k on the set M at a point p ∈ M .
Example 3.1.24 (Germs). Given the topological space
topology. Considering the functions f ≡ 0 and


0,
g(x) = −x − 1,


x − 1,
R
with the standard
t ∈ I[−1,1]
x < −1 .
x>1
(3.1.45)
The neighborhood U = I(−1,1) in the standard topology yields for every x ∈ U ,
This gives that the functions f and g belongs to the same equivalence class (for example C ∞ ) at for example p = 0 ∈ U .
f (x) = g(x).
Example 3.1.25 (Germs). Given the topological space R with the standard
topology. Considering the functions f (x) = x and g(x) = x3 . These functions
do not belong to the same germ because for every neighborhood U the functions
f and g do not coincide since f (x) = g(x) only at x ∈ {−1, 0, 1}.
Example 3.1.26 (Germs). Given the topological space R with the discrete topology. Considering the functions f (x) = x and g(x) = x3 . These functions belong
to the same germ in Cp0 (M ) at for example p = 0. Choose the the neighborhood
U = {0}. Here f (U ) = g(U ), so they belong to the same equivalence class at p.
Denition 3.1.18
(Linear map). A
all a, b ∈ R and f, g ∈ Cpk (M ):
linear map L : Cpk (M ) → R satises for
L(af + bg) = a · L(f ) + b · L(g)
(3.1.46)
Example 3.1.27 (Linear map). The function L(x) = cx where c ∈ R is a linear
map.
Consider: L(af + bg) = c(af + bg) = acf + bcg = a · L(f ) + b · L(g)
A derivation is simply dened as a linear map which satises the "product rule".
Where the name comes from observing that every derivation can be written as
a partial derivative as seen in the forthcoming Theorem 3.1.1.
Denition 3.1.19
on
Cpk (M )
(Derivation). A linear map L : Cpk (M ) → R is a
if for all f, g ∈ Cpk (M ):
L(f · g) = f (p) · L(g) + g(p) · L(f )
40
derivation
(3.1.47)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Example 3.1.28 (Derivation). Consider the linear map given by the directional
derivatives in the direction a = [a1 , . . . , an ]:
Da f =
n
X
i=1
ai
∂f
∂xi
(3.1.48)
Calculating Da (f g) at p gives:
Da (f g) =
n
P
i=1
g
ai ∂f
∂xi |p = f (p) ·
n
P
i=1
∂g
ai ∂x
|p + g(p) ·
i
n
P
i=1
∂f
ai ∂x
|p =
i
= f (p) · Da g + g(p) · Da f
From this follows that L is a derivation.
As already stated, an equivalent denition of tangent vectors are that they are
derivations. More formally a tangent vector is dened as:
Denition 3.1.20
(Tangent in abstract manifold). A
p ∈ M is a derivation on Cp∞ (M ).
tangent vector L at
Proof. See for example [BJ82] for a proof of the equivalence.
The main reason for providing this alternate denition is its usefulness in calculations which will be used now to prove that tangent spaces are vector spaces.
Before stating and proving the main theorem, the following lemma is needed.
Lemma 3.1.1. [Küh02] Let X be a tangent vector and f a constant function,
then
X(f ) = 0.
(3.1.49)
Proof. First assume f = 1. Using the denition of the derivation operation
(3.1.47):
X(1) = X(1 · 1) = 1 · X(1) + 1 · X(1) =
= 2 · X(1) =⇒ X(1) = 0
In the general case f has a constant value c and using the denition of linear
operator (3.1.46):
X(c) = X(c · 1) = c · X(1) = 0
41
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Theorem 3.1.1. [Küh02] For an n-dimensional manifold M , the tangent space
at p is an n-dimensional vector space of Rn and Tp M is spanned in any coordinate system x1 , ..., xn for a given chart by:
∂
∂
|p , ...,
|p
∂x1
∂xn
(3.1.50)
and for every tangent vector X at p:
X=
n
X
X(xi )
i=1
∂
|p
∂xi
(3.1.51)
Proof. Given a chart ϕ : U → V . Assume V = Bd (0, ) which can be done
without loosing generality and ϕ(p) = 0 so:
(3.1.52)
x1 (p) = ... = xn (p) = 0.
Let g : V → R be a smooth function and set f = g ◦ ϕ. Furthermore set:
Z
gi (y) =
0
1
∂g
(t · y)dt
∂ei
(3.1.53)
where ei is the i:th component of standard basis in Rn which satises:
(3.1.54)
xi (p) = ei (ϕ(p))
and also due to g being smooth gi is smooth. Now the following calculations:
n
P
∂g
∂ei (t · y)
i=1
= ∂g
∂t (t · y)
·
d(tei )
dt
= (where
d(tei )
dt
= ei )
implies the following using above calculation and (3.1.53):
n
P
n R
P
1
gi (y) · ei =
i=1
i=1
=
n
R1 P
0
i=1
∂g
∂ei (t
∂g
(t
0 ∂ei
· y) · ei dt =
· y)dt · ei =
R1
0
∂g
∂t (t
· y)dt = g(y) − g(0)
Now using the identities
f = g ◦ ϕ, fi = gi ◦ ϕ, xi = ei ◦ ϕ,
(3.1.55)
the above expression and using ϕ(p) = 0 gives:
f (q) − f (p) =
n
X
gi (ϕ(q)) · ei (ϕ(q)) + g(0) − g(ϕ(p)) =
i=1
n
X
i=1
42
fi (q) · xi (q) (3.1.56)
3.1 Manifolds
3 RIEMANNIAN GEOMETRY
Taking derivatives gives:
∂f
|p = fi (p)
∂xi
(3.1.57)
Given a tangent vector X at p the following comes from the denition of linear
map 3.1.18, tangent vector 3.1.20, (3.1.56) and Lemma 3.1.1:
X(f ) = X f (p) +
n
P
fi xi = X(f (p)) + X
n
P
fi xi =
i=1
i=1
by tangent vector of constant and linear map
=0+
n
P
X(fi xi ) =
i=1
by the denition of derivation 3.1.20
=
n
P
(xi (p) · X(fi ) + fi (p)X(xi ) =
i=1
by xi (p) = 0 and (3.1.57)
=
n
P
i=1
∂f
∂xi |p ·X(xi )
=
n
P
i=1
X(xi ) ·
∂
∂xi |p
(f )
for every f ∈ Cp∞ (M ) and left to show is that the vectors
independent and that the tangent space is a vector space.
∂
∂xi |p
are linearly
For Tp M to be a vector space it needs to satisfy the conditions in Denition
D.0.4. These conditions are very repetitive to prove so only that if
(3.1.58)
X1 , X2 ∈ Tp M, X1 + X2 ∈ Tp M
will be proven here and the rest will be omitted.
n
n
P
P
∂
X1 (f ) + X2 (f ) =
X1 (xi ) · ∂x
|
(f
)
+
X2 (xi ) ·
p
i
i=1
i=1
n
∂
P
|p (f ) ∈ Tp M
=
X1 (xi ) + X2 (xi ) · ∂x
i
i=1
43
∂
∂xi |p
(f ) =
3.2 Riemannian metrics
To show that
∂
∂xi |p
3 RIEMANNIAN GEOMETRY
is linearly independent lets consider the following equation:
(
1 ,i = j
∂
(3.1.59)
|p (xj ) =
∂xi
0 , i 6= j
∂
For ∂x
|p to be linearly independent the linear combination of vectors have to
i
satisfy:
X(f ) =
n
X
i=1
X(xi )
∂xj
|p (f ) = 0 ⇐⇒ X(xi ) = 0 for every i
∂xi
(3.1.60)
Assume that X(xi ) 6= 0 for some i. Then using 3.1.59 on
n
X
X(xi )
i=1
∂xj
|p (x1 )
∂xi
(3.1.61)
gives that X(x1 ) = 0 and continuing this for X(x2 ), X(x3 ),. . . , X(xn ) gives a
∂
|p is linearly independent.
contradiction and ∂x
i
3.2 Riemannian metrics
The second component in a Riemannian manifold is a Riemannian metric. Without any extra structure, the length of curves in a manifold is undened but given
a Riemannian metric, the length of curves and other geometric properties such
as angles will be dened. Once length of curves are dened, geodesics are also
dened. The Riemann metric denes at every point in the manifold an inner
product such that it varies smoothly when moving on the manifold, this gives
a family of inner products.
When dening Riemannian metrics, the concept of bilinear forms is needed. A
bilinear form is essentially a map which is linear in each of its two argument.
Denition 3.2.1
(Bilinear form). Given a vector space X and a eld (C.0.18
on page 108) of scalars, F . A bilinear form is a function B : X × X → F that
satises for all u, f, g ∈ X and c ∈ F :
a) B(u + f, g) = B(u, g) + B(f, g)
b) B(u, f + g) = B(u, f ) + B(u, g)
c) B(cf, g) = B(f, cg) = c · B(f, g)
Example 3.2.1 (Bilinear form). Given a vector space X over the eld R. This
vector space is equipped with an inner product h·, ·i. As seen for u, f, g ∈ X and
c ∈ R this function is a bilinear form.
a) hu + f, gi = hu, gi + hf, gi
b) hu, f + gi = hu, f i + hu, gi
44
3.2 Riemannian metrics
3 RIEMANNIAN GEOMETRY
c) hcf, gi = c · hf, gi = hf, cgi
Example 3.2.2 (Bilinear form). Given a vector space X over the eld C. This
vector space is equipped with an inner product h·, ·i. As seen for u, f, g ∈ X and
c ∈ C this function is not a bilinear form.
a0 ) hu + f, gi = hu, gi + hf, gi
b0 ) hu, f + gi = hu, f i + hu, gi
c0 ) hcf, gi = c · hf, gi = hf, cgi
In Denition 3.2.1 of bilinear forms condition a) and b) are fullled but not
A function which satises a) and b) and is conjugate linear as h·, ·i in this
example is called a sesquilinear form.
c).
The inner products will belongs to the space (L2 (Tp M ; R)) dened as:
Denition 3.2.2 ((L2 (Tp M ; R))).
This space is dened by:
L2 (Tp M ; R) = {f : Tp M × Tp M → R| f is bilinear}
(3.2.1)
L2 (Tp M ; R) has the basis:
{dxi |p ⊗ dxj |p | i, j = 1, ..n}
(3.2.2)
Here dxi is a dual basis in the dual space: (Tp M )∗
Where we dene:
∂
|p ) = δji =
dxi |p (
∂xj
(
1 , if i = j
0 , if i =
6 j
The bilinear forms dxi |p ⊗ dxj |p are dened by:
(
1 , if i = k and j = l
∂
∂
i j
|p,
|p = δk δl =
(dxi |p ⊗ dxj |p )
∂xk
∂xl
0 , otherwise
Inserting the basis and coecients for the representation gives:
X
α=
αij dxi ⊗ dxj
(3.2.3)
(3.2.4)
(3.2.5)
i,j
which gives the following expression:
∂
∂
αij = α
,
∂xi ∂xj
45
(3.2.6)
3.2 Riemannian metrics
3 RIEMANNIAN GEOMETRY
Then a Riemannian metric is a collection of α's as in (3.2.5) which are symmetric, positive denite and the coecients αij are dierentiable.
Denition 3.2.3 (Riemannian metric). On a given manifold M , a Riemannian
metric g is an association for points p ∈ M :
p → gp ∈ L2 (Tp M ; R)
(3.2.7)
such that the following is satised for all x, y ∈ Tp M :
a) gp (x, y) = gp (y, x) (Symmetry)
b) gp (x, x) > 0 for all x 6= 0 (Positive deniteness)
c) In every local representation (in every chart) the coecients gij
X
gp =
gij (p)dxi |p ⊗ dxj |p
(3.2.8)
i,j
are dierentiable functions. (Dierentiability)
Remark 3.2.1. The inner product dened by the Riemann metric g at a point
p ∈ T M is the gp from the above denition. If the condition of positive deniteness in Denition 3.2.3 is replaced by the condition that if:
gp (x, x) = 0 =⇒ x = y,
(3.2.9)
then g is a semi-Riemannian metric (or pseudo-Riemannian metric). A classical
example of a semi-Riemannian metric is the Lorentzian metric from the general
relativity theory.
Example 3.2.3 (Riemannian metric). Given an open set U ∈ Rn and let the
basis be given by e1 , . . . , en . Then using ∂x∂ = ei and (3.2.6)gives:
i
(3.2.10)
gij = gp ei , ej
If gp is the Euclidean inner product then gij is the identity matrix:

1
0

gij =  .
 ..
0
0
1
...
...
...
0
..
.
...

0
0

.
.. 
.
1
(3.2.11)
Due to gp being an inner product a) and b) in the denition of Riemannian
metric 3.2.3 is fullled. For gp to be a Riemannian metric the coecients gij
has to be dierentiable functions which is excluded here but satised and hence
the function g(·, ·) = h·, ·i is a Riemannian metric on U .
46
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
Now a Riemannian manifold is a manifold together with a Riemannian metric.
This object has the sucient structure for further analysis, the dierentiability
of the manifold make it possible to use calculus on complicated objects and the
Riemann metric gives length of curves which give rise to geodesics which in turn
are essential to this thesis.
Denition 3.2.4 (Riemannian manifold). A pair (M, g) is a Riemannian manifold if M is a manifold and g is a Riemannian metric
Example 3.2.4 (Riemannian manifold). The pair (U, g) where U is an open
set, U ∈ Rn and g is the Riemannian metric from Example 3.2.3 is a Riemannian manifold.
An important question is if there exist Riemann metrics for a given manifold
and if so whether they are unique. As proven in Theorem 3.2.1, there exist indeed Riemann metrics for any given manifold but they are generally not unique
as seen by the following example.
Example 3.2.5 (Non-uniqueness of Riemannian metric). Given an open set
In Example 3.2.3 a Riemann metric to this manifold was given by
the Euclidean inner product. Another example of a Riemann metric for this
manifold is given by:
U ∈ Rn .
(3.2.12)
gij (x1 , . . . , xn ) = δij (1 + xki xkj )
where k is an even integer and is given in matrix form by:

1 + xk1
 0

gij =  .
 ..
0
0
1 + xk2
...
...
0
0
...
0
1 + xkn
..
.
...
..
.



.

(3.2.13)
Theorem 3.2.1 (Existence). [GHL04] There exist at least one Riemannian
metric g on any given manifold M .
Proof. The proof is omitted here and can be found in [GHL04].
3.3 Riemannian connections
In Section 3.1 the problem of derivatives on abstract manifolds are dealt with
for the case of scalar functions by Denition 3.1.20 of tangent vectors. In this
section the notion of derivatives on abstract manifolds will be introduced by
dening the derivative of vector elds with respect to a tangent vector which
will give a tangent vector. One way to do this is by the Lie bracket which does
not use the Riemannian metric and another is the Riemannian connection which
uses the Riemannian metric and is a generalization of the covariant derivative
47
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
which is the usual derivative of vector elds with respect to a tangent vector
in dierential geometry. As will be shown the Riemannian connection is in fact
unique for every Riemannian metric and using the Riemann connection, the
Christoel symbols can be expressed which will then be used to express the
geodesic equation in local coordinates.
The Lie bracket or the Lie derivative is a measure of the non-commutativity of
the derivatives of the vector elds and is dened as:
Denition 3.3.1 (Lie bracket).
Given a Riemannian manifold (M, g) with vector elds X, Y on M and a smooth function f : M → R. The Lie bracket [X, Y ]
of X, Y is dened as:
[X, Y ](f ) = X(Y (f )) − Y (X(f ))
and at a point p the
(3.3.1)
Lie bracket is dened as:
[X, Y ]p (f ) = Xp (Y (f )) − Yp (X(f ))
(3.3.2)
Remark 3.3.1. Note that in the denition of Lie bracket, the Riemannian metric g was not used so only the dierential structure is sucient for a denition.
Also, the Lie bracket [X, Y ] is a vector eld which can be realized by remembering that the set of tangent vectors is a vector space.
Example 3.3.1 (Lie bracket). Given the vector elds X, Y , where
X = xy
∂
∂x
and
Y = (x + y)
(3.3.3)
∂
∂y
(3.3.4)
and the smooth function f : S1 → R, where f (x, y) = x + y. Then the Lie
bracket is given by:
∂
[X, Y ](f ) = xy
(x+y)
∂(x+y)
∂y
∂x
∂
− (x + y)
(xy)
∂(x+y)
∂x
∂y
=
= xy − (x2 + xy) = −x2
Below follows six important properties of Lie brackets which will be used continuously throughout this section when calculating with Lie brackets.
Proposition 3.3.1 (Properties of Lie brackets). [Küh02] Given a Riemannian
manifold M with vector elds X, Y, Z on M , constants a, b ∈ R and smooth
functions f, h : M → R. The Lie bracket satises the following properties:
48
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
a) [aX + bY, Z] = a[X, Z] + b[Y, Z] (Linear in rst argument)
b) [X, Y ] = −[Y, X] (Anti-symetric)
c) [f X, hY ] = f · h · [X, Y ] + f · X(h) · Y − h · Y (f ) · X
d) X, [Y, Z] + Y, [Z, X] + Z, [X, Y ] = 0 (Jacobi identity)
e) [ ∂x∂ i , ∂x∂ j ] = 0
for every chart with coordinates [x1 , . . . , xn ]
f)
P
i
∂
,
ξi ∂x
i
dinates)
P
j
∂
ηj ∂x
j
P
∂ηj
∂ξj
∂
=
ξi ∂xi − ηi ∂xi ∂x
j
i,j
(Representation in local coor-
Proof.
a): [aX + bY, Z](f ) =
= a · X(Z(f )) + b · Y (Z(f )) − Z(a · X(f ) + b · Y (f )) = (Z is a linear map)
= a · X(Z(f )) − a · Z(X(f )) + b · Y (Z(f )) − b · Z(Y (f )) =
= a · [X, Z](f ) + b · [Y, Z](f )
b): [X, Y ](f ) = X(Y (f )) − Y (X(f )) = − Y (X(f )) − X(Y (f )) = −[Y, X](f )
c): [f X, hY ](θ) = f ·X(h·Y (θ))−h·Y (f ·X(θ)) = (using equation (3.1.47) in Denition 3.1.19)
= f · X(h)Y (θ) + f · h · X(Y (θ)) − h · Y (f )(X(θ)) − h · f · Y (X(θ)) =
= (f · h · [X, Y ] + f · X(h) · Y − h · Y (f ) · X)(θ)
for every function θ : M → R .
d): Using:
X,[Y, Z] (f ) = X [Y, Z](f) −[Y,
Z](X(f )) =
= X Y (Z(f ) −Z Y (f ) − Y Z(X(f )) −Z Y (X(f ))
gives:
( X,
[Y, Z] + Y, [Z, X] + Z,
[X,
Y ] )(f ) =
= X Y (Z(f ) −Z Y (f ) − Y Z(X(f )) −Z Y (X(f )) +
+ Y Z(X(f ) −X Z(f ) − Z X(Y (f )) −X Z(Y (f )) +
+ Z X(Y (f ) −Y X(f ) − X Y (Z(f )) −Y X(Z(f )) =
e):
=0
∂
∂
∂xi , ∂xj
=
2
∂ f
∂xi ∂xj
(f ) =
−
2
∂
∂xi
∂ f
∂xj ∂xi
∂
∂xj (f )
∂
− ∂x
j
∂
∂xi (f )
=0
49
=
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
f): Considering:
X
i
∂ X
∂
ξi
,
ηi
(xj )
∂xi i
∂xi
(3.3.5)
gives:
P ∂ P ∂xj
P
P ∂xj
P ∂ P
∂
∂
(x
)
=
−
ξi ∂xi , ηi ∂x
ξ
η
η
ξ
j
i ∂xi
i ∂xi
i ∂xi
i ∂xi =
i
i
i
i
i
i
i
P ∂ξ
P ∂η
= ξi ∂xji − ηi ∂xji
i
i
Using (3.1.51) gives that:
X
X
∂ X
∂
∂ηj
∂ξj
∂
ξi
,
ηj
=
ξi
− ηi
∂x
∂x
∂x
∂x
∂x
i
j
i
i
j
i
j
i,j
(3.3.6)
The meaning of the term Riemannian connection (or Levi-Civita connection)
lies in the "connection" between the tangent spaces which by denition are disjoint. These tangent spaces at dierent points in the manifold are related by
the Riemannian connection as following:
Denition 3.3.2 (Riemannian connection).
Given a Riemannian manifold (M, g).
A Riemannian connection ∇ on (M, g) is a map:
∇ : TM × TM → TM
(3.3.7)
which associate two vector elds X and Y with a vector eld ∇X Y such that
the following is satised for a smooth function f : M → R:
a) ∇X1 +X2 Y = ∇X1 Y + ∇X2 Y (additive in the subscript)
b) ∇f X Y = f · ∇X Y (linear in the subscript)
c) ∇X (Y1 + Y2 ) = ∇X Y1 + ∇X Y2 (additive in the argument)
d) ∇X (f Y ) = f · ∇X Y + (X(f )) · Y (product rule in the argument)
e) X(g(Y, Z)) = g(∇X Y, Z)+g(Y, ∇X Z) (relation with the Riemannian metric)
f) ∇X Y − ∇Y X = [X, Y ] (torsion-free)
Example 3.3.2 (Riemann connection). Given the Riemannian manifold (Rn , g),
where g is the Euclidean inner product. For vector elds:
X = [X1 , . . . , Xn ] =
X
i
ai
X
X
∂
∂
∂
, Y =
bi
, Z=
ci
,
∂xi
∂xi
∂xi
i
i
(3.3.8)
the directional derivative:
DX Y =
X
i
Xi
∂Yi ∂
= ∇X Y
∂xi ∂xi
is a Riemannian connection on (Rn , g) which is shown below:
50
(3.3.9)
3.3 Riemannian connections
a): ∇X+Z Y =
b): ∇f X Y =
P ∂bi P ∂bi
P
∂bi
=
(ai + ci ) ∂x
ai ∂xi + ci ∂xi = ∇X Y + ∇Z Y
i
i
P
i
3 RIEMANNIAN GEOMETRY
i
∂bi
f ai ∂x
i
=f
P
i
c): ∇X (Y + Z) =
P
d): ∇X (f Y ) =
bi )
ai ∂(f
∂xi = f ·
P
i
i
i
∂bi
ai ∂x
i
i +ci )
=
ai ∂(b∂x
i
P
P
i
i
= f · ∇X Y
∂bi
+
ai ∂x
i
∂bi
ai ∂x
+
i
P
P
i
i
∂ci
= ∇X Y + ∇X Z
ai ∂x
i
∂f
ai ∂x
bi = f · ∇X Y + (X(f )) · Y
i
Properties e) and f ) are omitted here but can be computed in a similar manner although more tedious.
As already stated, for a given Riemannian manifold, there exists an unique Riemannian connection which is formalized in the following theorem.
Theorem 3.3.1. [Küh02] [KN63] Given a Riemannian manifold (M, g). Then
there exists an unique Riemannian connection ∇ on (M, g).
Proof. First prove the uniqueness of the connection ∇. For vector elds X, Y, Z ,
the following three equalities holds true using the property of the relation with
the Riemannian metric in the denition of the Riemannian connection 3.3.2 and
denoting gp (·, ·) = h·, ·i.
XhY, Zi = h∇X Y, Zi + hY, ∇X Zi
(3.3.10)
Y hX, Zi = h∇Y X, Zi + hX, ∇Y Zi
(3.3.11)
− ZhX, Y i = −h∇Z X, Y i − hX, ∇Z Y i
(3.3.12)
Adding (3.3.10), (3.3.11) and (3.3.12) gives:
XhY, Zi + Y hX, Zi − ZhX, Y i =
= hY, ∇X Z−∇Z Xi+hX, ∇Y Z−∇Z Y i+hZ, ∇X Y +∇Y Xi = (∇X Y −∇Y X = [X, Y ])
= hY, [X, Z]i + hX, [Y, Z]i + hZ, 2∇Y X + [Y, X]i
Rearranging terms in the above equation gives the Koszul formula below:
2h∇X Y, Zi = XhY, Zi+Y hX, Zi−ZhX, Y i−hY, [X, Z]i−hX, [Y, Z]i−hZ, [Y, X]i
(3.3.13)
Given Z , the right-hand side of (3.3.13) is uniquely determined. Assume that
there exists an U 6= ∇X Y such that for all Z :
h∇X Y, Zi = hU, Zi.
(3.3.14)
h∇X Y − U, Zi = 0
(3.3.15)
Then
51
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
and this is especially true for
Z = ∇X Y − U
(3.3.16)
h∇X Y − U, ∇X Y − U i = 0,
(3.3.17)
which gives that:
using that M is Hausdor and h·, ·i is an inner product together with equation
(3.3.17) gives ∇X Y = U and hence ∇X Y is unique.
Now to show the existence of ∇, dene it as satisfying the Koszul formula
(3.3.13) for every X, Y, Z ∈ Γ(T M ).
Remain to show is that ∇X Y is well-dened and thus:
h∇X Y |p , Zp i
(3.3.18)
only depends on Zp .
For ∇ to be a Riemannian connection it has to satisfy condition a)-f ) in Denition 3.3.2.
a) Using (3.3.13) on 2h∇X1 +X2 Y, Zi − 2h∇X1 Y + ∇X2 Y, Zi gives:
2h∇X1 +X2 Y, Zi − 2h∇X1 Y + ∇X2 Y, Zi = (X1 + X2 )hY, Zi + Y hX1 + X2 , Zi −
− ZhX1 + X2 , Y i − hY, [X1 + X2 , Z]i − hX1 + X2 , [Y, Z]i − hZ, [Y, X1 + X2 ]i −
− (X1 hY, Zi + X2 hY, Zi + Y h(X1 , Zi + Y h(X2 , Zi − ZhX1 , Y i − ZhX2 , Y i −
−hY, [X1 , Z]i−hY, [X2 , Z]i−hX1 , [Y, Z]i−hX2 , [Y, Z]i−hZ, [Y, X1 ]i−hZ, [Y, X2 ]i =
=0
by using the bilinearity of the inner product, the anti-symmetry of the Lie
brackets and collecting terms.
b) Using (3.3.13) on 2h∇f X Y, Zi − 2hf ∇X Y, Zi gives:
(f XhY, Zi+Y hf X, Zi−Zhf X, Y i−hY, [f X, Z]i−hf X, [Y, Z]i−hZ, [Y, f X]i)−
−(f XhY, Zi+Y hX, Zi−f ZhX, Y i−f hY, [X, Z]i−f hX, [Y, Z]i−f hZ, [Y, X]i) =
= [ Using the product rule for Lie brackets and the bilinearity of inner products ] =
= Y f hX, Zi−Zf hX, Y i+f ZhX, Y i−f Y hX, Zi−hY, f [X, Z]i−hY, −(Zf )Xi+
52
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
+ f hY, [X, Z]i − hZ, −f [X, Y ]i − hZ, −(Y f )Xi + f hZ, [Y, X]i =
= [ Using the product rule on Zf hX, Y i = (Zf )hX, Y i+f ZhX, Y i and Y f hX, Y i =
= (Y f )hX, Zi + f Y hX, Zi]
= (Y f )hX, Zi + f Y hX, Zi − (Zf )hX, Y i − f ZhX, Y i − f Y hX, Zi +
+ f ZhX, Y i + (Zf )hX, Zi − (Y f )hX, Zi =
= 0.
The rest of the conditions c) − f ) are done in a similar manner and are
skipped in this thesis.
Now the goal with the rest of this section is to express an equivalent denition
of geodesics in the Riemannian geometry to and use this denition to nd an
equation in local coordinates for the geodesic equation. The Christoel symbols
are expressed in terms of the Riemann metric and the Riemann connection.
The new denition of geodesics gives an equation a curve has to satisfy to be a
geodesic. When expressing this in local coordinates, the Christoel symbols are
expressed and more generally the Christoel symbols arises when dealing with
Riemann connections in local coordinates.
Denition 3.3.3 (Christoel symbols). The Christoel symbols of the rst kind
Γij,k is dened as:
Γij,k =
and the
∂
1
∂
∂
(−
gij +
gik +
gjk )
2 ∂xk
∂xj
∂xi
(3.3.19)
Christoel symbols of the second kind Γm
ij is dened as:
Γm
ij =
X
Γij,k g km , ∇
k
we then have:
∇
∂
∂xi
∂
∂xi
X
∂
∂
=
Γkij
∂xj
∂xk
(3.3.20)
k
X
∂
∂
=
Γkij
∂xj
∂xk
(3.3.21)
k
The following proposition gives a formula for the Riemann connection in local
coordinates.
Proposition 3.3.2. Given vector elds:
X=
X
i
ξi
X
∂
∂
, Y =
ηj
∂xi
∂x
j
j
53
(3.3.22)
3.3 Riemannian connections
3 RIEMANNIAN GEOMETRY
in local coordinates. Then ∇X Y in local coordinates is given by:
∇X Y =
XX
k
ξi
i
∂ηk X k
∂
+
Γij ξi ηj
.
∂xi
∂x
k
i,j
(3.3.23)
Proof. Using the propertiesa)-d) in Denition
3.3.2
on ∇X Y gives:
∇X Y = ∇P ξ i
i
= ξ1 η 1 ∇
∂
∂x1
∂
∂xi
P
∂
= ξ1 ∇
ηj ∂x
j
j
∂η1 ∂
∂
+
ξ
1
∂x1
∂x1 ∂x1
∂η1 ∂
∂
∂x1 + ξ2 ∂x2 ∂x1
∂
∂x1
P
j
+ . . . + ξ1 η n ∇
ηj ∂∂j + . . . + ξn ∇
∂
∂x1
+ ξ2 η1 ∇ ∂
+ . . . + ξn ηn ∇ ∂
∂xn
∂x2
P P ∂ηk P k
∂
=
ξi ∂xi + Γij ξi ηj ∂xk
k
i
∂
∂xn
∂
∂xn
∂
n
+ ξ1 ∂η
∂x1 ∂xn +
∂
∂xn
∂ηn ∂
+ ξn ∂x
=
n ∂xn
P
j
ηj ∂∂j =
i,j
An alternative equivalent denition of geodesics 2.3.4 in Chapter 2 for Riemannian geometry is given below.
Denition 3.3.4 (Geodesic in Riemannian geometry). Given a Riemannian
manifold (M, g) and a curve γ . The curve γ is a geodesic in a Riemannian
geometry if
dγ
dγ
=λ
(3.3.24)
∇ dγ
dt dt
dt
for some function λ : M → R and if γ is a natural parametrization (see Denition
2.2.4),
dγ
∇ dγ
=0
(3.3.25)
dt dt
To see why this equation make sense at least in E n as a denition for geodesics,
consider the equation:
dγ
dγ 2
= 2.
(3.3.26)
∇ dγ
dt dt
dt
This is the vector of acceleration for a particle moving along a curve γ and if
the particle moves without the inuence of acceleration, it will follow a straight
line which are the geodesics in E n and without acceleration (3.3.24) is reduced
to a geodesic equation. If instead γ is a curve on S2 . Then the vector of acceleration is given by: ∇ dγ dγ
. The geodesics will be the path particles follow
dt dt
without the inuence of acceleration which will be the great circles which again
coincide with what was discovered in Chapter 2.
Proof. (Equivalence of the denitions) The proof of the equivalence of the two
denitions is omitted here but can be found in [Jos11].
54
3.4 Geodesics
3 RIEMANNIAN GEOMETRY
Using Proposition 3.3.2 together with (3.3.25) gives a system of dierential equations for the geodesic equation in local coordinates.
Proposition 3.3.3. [GHL04] Given a Riemannian manifold (M, g). Then a
naturally parametrized curve is a geodesic if it is a solution to the following differential equations in local coordinates:
d2 xk X k dxi dxj
+
Γij
= 0,
dt2
dt dt
i,j
(3.3.27)
where xi are the coordinates for γ .
Proof. See [GHL04] for a proof of this proposition.
3.4 Geodesics
In Chapter 2, which concerns metric geometry, several results about when shortest paths exist in length spaces was given. To use these results for a Riemannian
manifold, a suitable metric in the Riemannian geometry is needed in order for
a Riemannian manifold to be a length space. This can and will in fact be done
later in this subsection and as will be seen the choice of metric is logical and
straightforward.
As stated in the introduction to this chapter, the Riemannian metric is used
when dening the length of a curve by "weighting" the dierent parts of the
path of the curve.
Denition 3.4.1 (Length of curve). Given a Riemannian manifold (M, g) and
a continuously dierentiable curve γ : I[a,b]→M on M . The length of the curve
l(γ) is given by:
l(γ) =
Zb q
gγ(t) (γ 0 (t), γ 0 (t))dt
(3.4.1)
a
Example 3.4.1 (Length of curve). Given a Riemannian manifold (M, g), where
g is the Euclidean inner product. The length of the curve:
(
γ(t) =
x = t,
y = t,
t ∈ I[0,1]
t ∈ I[0,1]
(3.4.2)
is given by:
l(γ) =
√
R1 p
R1 p
R1 √
gγ(t) (γ 0 (t), γ 0 (t))dt =
h[1, 1], [1, 1]idt =
2dt = 2.
0
0
0
The goal is to choose a metric such that the corresponding metric space is a
length space. The metric then needs to coincide with the intrinsic metric from
55
3.4 Geodesics
3 RIEMANNIAN GEOMETRY
Chapter 2 in order to be a length space. Because length of curves are only
dened for C 1 curves, the inmum of every path of C 1 curves between the given
points will coincide with the intrinsic metric because the manifolds are smooth.
Denition 3.4.2 (Metric on Riemannian manifold).
Riemannian manifold (M, g) is given by:
A metric d on a connected
d(p1 , p2 ) = inf l(Φ)
(3.4.3)
where Φ is the set of C 1 curves of the form γ : I[a,b] → M,
γ(a) = p1 and γ(b) = p2 .
Left to show for the above dened metric to be a length space is that it is indeed
a metric and that the topology generated by this metric coincides with the usual
topology of the manifold.
Proposition 3.4.1. [GHL04] The object d dened in Denition 3.4.2 is a
metric and the topology generated by the metric is the original topology of the
manifold.
Proof. First prove that d is a metric. In order to be well-dened, d has to be
dened for every pair x, y ∈ M .
Choose x ∈ M and let Ux be the set of points joined to x by piecewise C 1
curves. Then Ux is an open set and
[
U \ Ux =
Uy
(3.4.4)
y ∈U
/ x
is hence an open set because it is an union of open sets. Now Ux is non-empty
because x ∈ Ux and U \ Ux is the empty set, due to otherwise there would
be two disjoint open sets Ux , U \ Ux which covers M and hence M would be
disconnected. Then Ux = M and d is well-dened.
A metric satises:
d(x, y) = d(y, x).
(3.4.5)
Because length of curves is independent of backward and forward parametrization of curves this is fullled. The triangle inequality,
d(x, y) ≤ d(x, z) + d(z, y)
(3.4.6)
comes from the fact that the distance is the inmum of the length of the curves
and hence the length of a piece-wise C 1 curve between x to y passing z cannot
be greater than d(x, y).
The last property d must satisfy is:
d(x, y) = 0 ⇐⇒ x = y.
56
(3.4.7)
3.4 Geodesics
3 RIEMANNIAN GEOMETRY
If x = y , then clearly d(x, y) = 0. Now prove the converse assertion. Assume
x 6= y . Now there exists a chart (X, f ) around x such that:
y∈
/ X, f (x) = 0, f (X) = Bh (0, 1)
(3.4.8)
because M is Hausdor and h is the metric induced by f on Bh (0, 1) by the
Riemannian metric g i.e. the "Riemannian metric" on the submanifold X . Let
k·k denote the Euclidean norm in Rn . Using the compactness of Bh (0, 21 ) gives
that there exist λ, µ > 0 such that for a given p ∈ Bh (0, 12 ) and u ∈ Tp M :
λkuk2 ≤ hp (u, u) ≤ µkuk2 .
(3.4.9)
Choosing a curve γ between x and y and calculating l(γ) (see [GHL04]) gives
that l(γ) > 0, which in turn gives: d(x, y) > 0 which is a contradiction and
hence d is a metric.
What is left to prove is that the topologies coincide which is omitted here and
can be found in [GHL04].
Summarizing the above results. A connected Riemannian manifold together
with the metric dened in Denition 3.4.2 is a length space. Together with the
fact from Proposition 3.4.1 that this metric induces the same topology as the
original topology of M gives that a connected Riemannian manifold is a length
space and the theory in Chapter 2 about length spaces is applicable. Rephrasing the Hopf-Rinow-Cohn-Vossen Theorem 2.3.3 for Riemannian manifolds gives
the following:
Theorem 3.4.1. Given a locally compact theorem Riemannian manifold
Then the following conditions are equivalent
M.
a) M is a complete space.
b) M is a boundedly compact space.
and if the conditions a) and b) are fullled, for every x, y ∈ M such that
d(x, y) < ∞ there is a shortest path between x and y
Proof. This theorem follows easily from Theorem 2.3.2, Hopf-Rinow-Cohn-Vossen
Theorem 2.3.3 and that M is a length space.
Example 3.4.2 (Existence of shortest paths). Given a Riemannian manifold
(S2 , g) with a Riemannian metric g . This manifold is locally compact, complete
and there exist rectiable paths between every point. Then there exist shortest
paths (not necessarily unique) between every pair of points in S2 .
57
4 FINSLER GEOMETRY
4
Finsler geometry
The Finsler geometry is a generalization of the Riemannian geometry. In the
Riemannian geometry, the Riemannian metric induces an inner product but the
function corresponding to the Riemann metric in the Finsler geometry induces
something called a Minkowski norm instead of an inner product. One other way
the Minkowski norms dier is that they are functions on the tangent bundle and
hence they have dependence of both the position in the manifold and the corresponding tangent space. Reusing the example of the hiker in the mountainous
region, used in the Riemannian geometry. In the Finsler geometry the length of
the path the hiker traverses is a function of which Minkowski norm is used and
hence the length of the path does not only depends on which places were visited
but also in which direction the hiker traveled at the time, so for example a path
downwards a mountain may be shorter than the same path traveled upwards
the mountain in the Finsler geometry.
The Finsler geometry is named after Paul Finsler who studied it in his dissertation from 1918. The name was phrased by Élie Cartan who was also one of the
early pioneers in this subject in the rst part of the 20:th century. Among other
important contributers in this subject are: Ludwig Berwald, Vagner Busemann,
Makoto Matsumoto and Shiing-Shen Chern. Most of the result in this chapter
can be found in [BCS00] and for those not familiar with dierential forms there
is a section in Appendix D covering most of what is needed as a prerequisite
about dierential forms to understand this chapter and for those interested in
a more thorough explanation of dierential forms the book [Car06] is recommended.
The content of this chapter is two sections covering the classical approach to
Finsler geometry which follows the approach in Riemannian geometry closely by
which introducing Finsler manifolds and the Chern connection which replaces
the Riemannian connection in the in Riemannian geometry. Then comes a section covering geodesics in the Finsler geometry where it will be shown that the
approach used to create a length space by introducing a suitable metric in the
Riemannian geometry will fail in the Finsler geometry.
4.1 Finsler manifolds
The Finsler manifold is a generalization of the Riemannian manifold. As with
the Riemannian manifold, it is a manifold together with a function which decides
the geometrical properties of the Finsler geometry such as angles and length of
curves.
In this chapter let x denote a point in a manifold M and let y ∈ Tx M if not
otherwise specied. Furthermore let the map:
π : T M → M, π(x, y) = x
(4.1.1)
be the natural projection (see Figure 10). By the object T M \ {0} means every
element of T M except when y = 0.
59
4.1 Finsler manifolds
4 FINSLER GEOMETRY
Denition 4.1.1 (Finsler manifold). Given a manifold M . A Finsler structure
F is a function F : T M → R+ which satises for all x, y ∈ T M and real λ > 0:
a) F is smooth on T M \ {0} (Regularity)
b) F (x, λy) = λF (x, y) (Positive homogeneity)
c) The n × n matrix:
1 ∂2
gij =
F2
2 ∂yi ∂yj
(4.1.2)
is positive-denite at every point of T M \ {0}
A pair (M, F ) is a
Finsler manifold.
Remark 4.1.1. The function F in Denition 4.1.1 evaluated at a specic tangent
space is a Minkowski norm and as can be proven it is in fact a norm as seen
in [BCS00].
Example 4.1.1 (Finsler manifold). Given a manifold
metric g. The function F , dened in a point x ∈ M as:
F (x, y) =
p
M
and a Riemannian
gx (y, y)
(4.1.3)
is a Finsler structure. Due to the smoothness of gij , the function F is a smooth
function because F is positive on T M \ {0}. Using that gx is an inner product
yields that:
F (x, λy) =
p
p
gx (λy, λy) = λ gx (y, y) = λF (x, y)
(4.1.4)
for λ > 0. The positive deniteness of
1 ∂2
F2
2 ∂yi ∂yj
(4.1.5)
is due to the Riemannian metric gx at a point x being positive-denite.
Remark 4.1.2. A Finsler manifold which is also a Riemannian manifold is a
Riemannian Finsler manifold .
Example 4.1.2 (Finsler manifold). For a given µ > 0 and y = [y1 , y2 ]. Let the
Finsler structure F (x, y) be independent of x and given by:
F (x, y) =
rq
y14 + y24 + µ(y12 + y22 )
(4.1.6)
Then the pair (R2 , F ) is a Finsler manifold. These kind of Finsler manifolds
where F is independent of x are known as locally Minkowskian manifolds. The
60
4.2 Connections
4 FINSLER GEOMETRY
function F is a smooth function on T M \ {0} and calculating F (λy1 , λy2 ) for
λ > 0 gives:
F (λy1 , λy2 ) =
rq
(λy14 ) + (λy2 )4 + µ((λy1 )2 + (λy2 )2 ) = λF (y1 , y2 ) (4.1.7)
Using the denition and calculating gij gives:
g11
g21
g12
g22

λ+

=
y12 (y14 +3y24 )
(y14 +y24 )3/2
−2y13 y23
(y14 +y24 )3/2
−2y13 y23
(y14 +y24 )3/2
λ+
y22 (y24 +3y14 )
(y14 +y24 )3/2

(4.1.8)

.
Observing that (4.1.8) is an Hermitian matrix and that the determinant:
g11
det
g21
g12
g22
>0
if λ > 0 by Example C.1.1 and using Theorem C.1.1 gives that
positive-denite.
(4.1.9)
g11
g21
g12
g22
is
From now on in this chapter a variation of the Einstein summation convention
will be used, i.e. expressions of the type:
X=
n
X
Xi ei
(4.1.10)
i=1
is written as X = Xi ei with the Einstein summation convention. Vectors will
have superscript indices v i where i goes from 1 to n, basis elements ci will have
subscript indices and repeated indices in a expression in both superscript and
subscript means summing over that index. The indices of summation are assumed to be between
1 and n. This gives that a vector v can be written as:
v = ci ei where ei is a basis and ei is a vector.
For covectors (elements of the dual space) the indices are subscript and the
elements of the basis are superscript indices. This gives that a covector w can
be written as: w = ci e∗i where {e∗i } is a basis of the dual space and e∗i is a
covector. For lowering and raising indices the object gij and its inverse g ij is
used respectively.
4.2 Connections
In the Riemannian geometry, the Riemannian connection was used to measure
the derivative of vector elds with respect to a tangent vector and thus this
gives a "connection" between dierent tangent spaces. In the Finsler geometry
several dierent connections arises. In this thesis the following will be used:
Non-linear connection, Ehresmann connection, linear connection and the Chern
connection. The Chern connection will have the same role as the Riemannian
connection in the Riemannian geometry and the goal of the theory of this chapter is to dene and prove that the Chern connection always exists and is unique
for a given manifold. The meaning of the term connection can informally be
61
4.2 Connections
4 FINSLER GEOMETRY
thought of as an object which can connect points in dierent charts which otherwise would not easily be compared. For a more formal treatment on connection
with emphasize on Finsler geometry the book [Mat86] is recommended.
Before the Chern connection can be dened a lot of ground work is needed.
Several abstract object and spaces will be presented below in order to ease the
calculation later on.
Denition 4.2.1
(Sphere bundle). Given a manifold (M ). The
SM is the set of points:
y
x,
∈ T M \ {0} ,
kyk
sphere bundle
(4.2.1)
y
where an element [x, y] is identied in the sphere bundle by: x, kyk .
Example 4.2.1 (Sphere bundle). Given the Riemannian Finsler manifold (S1 , g),
where g is the Euclidean inner product. Then T M is given by S1 × R from Example 3.1.22 and hence SM is the set of points:
[x, y] : x ∈ S1 , y =
±[x2 − x1 ]
k[x2 − x1 ]k
(4.2.2)
for x = [x1 , x2 ].
Figure 10: Tangent bundle
The method used for dening the sphere bundle can also been seen as treating
every ray:
{[x, λy] : λ > 0}
(4.2.3)
62
4.2 Connections
4 FINSLER GEOMETRY
as a single point and dene Tx M in this point (see Figure 4.2.3) together with
the inner product:
gij (x, y)dxi ⊗ dxj .
(4.2.4)
The point of this is that gij (x, y) is invariant under rescaling, y → λy for λ > 0,
i.e.
gij (x, y) = gij (x, λy).
(4.2.5)
This shows that while the dimension of the vector bundle T M is 2n the dimension of SM is 2n − 1. Further on in this chapter other objects which are
invariant under rescaling will be considered to simplify the calculations.
The
canonical projection map
p : SM → M, p(x, y) = x
(4.2.6)
helps dene the pulled-back bundle p∗ T M . Another pulled-back bundle is π ∗ T M
on T M \ {0} which serves the same purpose as p∗ T M does on SM . This means
that π ∗ T M is a vector bundle over T M \ {0} and π ∗ T M at a specic point
(x, y) is a copy of Tx M .
On π ∗ T M there is a natural Riemannian metric
g = gij dxi ⊗ dxj
(4.2.7)
which is called the fundamental tensor and another important tensor is the Cartan tensor A dened by:
A = Aijk dxi ⊗ dxj ⊗ dxk ,
(4.2.8)
where
Aijk =
F ∂gij
F
∂3
=
F2
2 ∂y k
4 ∂y i ∂y j ∂y k
(4.2.9)
is invariant under rescaling and nally the object:
Cijk =
1
Aijk .
F
(4.2.10)
In the literature this tensor is an alternative denition of the Cartan tensor but
Cijk is not invariant under rescaling.
As in the Riemannian geometry, Christoel symbol for the Finsler geometry can
be dened as:
Denition 4.2.2 (Christoel symbols). In the Finsler geometry the formal
Christoel symbols of the second kind Γijk on T M \ {0} is given by:
1 ∂gsj
∂gjk
∂gks
Γijk = g is ( k −
+
)
2 ∂x
∂xs
∂xj
63
(4.2.11)
4.2 Connections
4 FINSLER GEOMETRY
Denition 4.2.3 (Non-linear connection).
The object:
(4.2.12)
i
Nji = Γijk y k − Cjk
Γkrs y r y s
on T M \ {0} is the
non-linear connection.
Because objects which are invariant under rescaling is preferable, introducing
the object:
Nji
= Γijk `k − Aijk Γkjk `r `s ,
F
where `i =
yi
F
denes the
(4.2.13)
distinguished section ` of π∗ T M by:
`=
yi ∂
∂
= `i i
F ∂xi
∂x
(4.2.14)
which gives an object which is invariant under rescaling and its dual analog on
π ∗ T ∗ M is the Hilbert form ω given by:
ω=
∂F i
dx .
∂y i
(4.2.15)
Several of the following computations are excluded in this thesis but can be
found in [Kaw56] and [Kik62].
A local change of variables xi = xi (x̃1 , . . . , x̃n ) for
Ñqp
F̃
=
Nji
F
gives:
∂ x̃p ∂ 2 xi ˜s
∂ x̃p ∂xj Nji
+
` .
∂xi ∂ x̃p F
∂xi ∂ x̃q ∂ x̃s
(4.2.16)
To see where the name non-linear connection comes from consider local coordi∂
∂
∂
nate changes on the coordinate basis ∂x
i and ∂y i . The transformation on ∂xi
is given by the slightly complicated expression:
∂
∂xi
∂ 2 xi q ∂
=
+
ỹ
p
p
∂ x̃
∂ x̃
∂ x̃p ∂ x̃q ∂y i
while the expression for variable changes on
∂
∂y i
∂
∂xi ∂
=
.
∂ x̃p
∂ x̃p ∂xi
(4.2.17)
is less complicated.
(4.2.18)
and the local coordinate basis dxi , dxj of the dual space of T M called the
cotangent space , is transformed by coordinate changes to:
dx̃p =
∂ x̃p i
dx
∂xi
64
(4.2.19)
4.2 Connections
4 FINSLER GEOMETRY
dỹ p =
∂ ỹ p i
∂ 2 x̃p
dy +
i
∂y
∂xi ∂xj
(4.2.20)
where the expression for dỹ p is in this case more simple than that of dx̃p .
Now replace
∂
∂xi
by:
δ
∂
∂
=
− Nji i
j
j
δx
∂x
∂y
(4.2.21)
δy i = dy i + Nji dxj
(4.2.22)
and dyi by:
but this object is not invariant under rescaling so instead introduce the following
object:
δy i
1
= dy i + Nji dxj .
F
F
(4.2.23)
i
j
Now note that the corresponding dual objects of δxδ j , δy
F are respectively dx ,
∂
F ∂y
i and that the above objects have easy transformations under coordinate
changes.
Now the following object on T M \ {0} is a Riemannian metric and known as
the Sasaki metric :
δy i
δy i
⊗
.
(4.2.24)
F
F
δ Using this metric, the space spanned
n i by
o δxj is the horizontal subspace and
the orthogonal space spanned by δy
is the vertical subspace . These spaces
F
build up what is known as an Ehresmann connection and this connection is
characterized by the quantities Nji which is where the name non-linear connection comes from.
gij dxi ⊗ dxj + gij
When measuring the rate of change of a tensor eld Y in the direction X at a
point p it has to satisfy the product rule. Let Λ be such an operator and let it
satisfy the product in the following way.
∂
∂
+ `j Λ X j ,
(4.2.25)
∂xj
∂x
∂
ΛX g = (dgij )(X)dxi ⊗ dxj gij (ΛX dxi ) ⊗ dxj + gij dxi ⊗ (ΛX dxj ). (4.2.26)
+
ΛX ` = (d`j )(X)
and dene the
covariant derivatives ΛX ∂x∂ j and ΛX dxi as:
∂
∂
= ωji (X) i ,
∂xj
∂x
ΛX dxi = −ωji (X)dxj .
ΛX
65
(4.2.27)
(4.2.28)
4.3 Geodesics
4 FINSLER GEOMETRY
In the Riemannian geometry, the Riemannian connection is a generalization
of the covariant derivative. Further generalizations of the covariate derivative
to the Finsler geometry gives the linear connections which are dened below.
Comparing these connections to the Riemannian geometry, the linear connections does not satisfy the torsion free condition which expressed in terms of
Finsler geometry is stated as:
d(dxi ) − dxj ∧ ωji = −dxj ∧ ωji = 0
(4.2.29)
Denition 4.2.4 (Linear connection).
The operator Λ on π ∗ T M which satises
the equations (4.2.25)-(4.2.28) together with the following conditions for vector
elds X, Y , scalar functions f and constants λ is a linear connection.
a) ΛX (f Y ) = (df )(X)Y + f ΛX Y
b) ΛX (Y1 + Y2 ) = ΛX Y1 + ΛX Y2
c) ΛλX Y = λΛX Y
d) ΛX1 +X2 Y = ΛX1 Y + ΛX2 Y
For some examples of dierent linear connection and other kind of connection,
the text book [Mat86] is recommended.
The linear connections are not expressed in terms of the Finsler structure. The
Chern connection is a linear connection which imposes constraints such that
it both relates to the Finsler structure and is torsion free. By this the Chern
connection generalizes the Riemannian connection. In the following theorem
the Chern connection is dened and as stated, it exists and is unique for a given
Finsler manifold as is the case in the Riemannian geometry for Riemannian
manifolds and the Riemannian connection.
Theorem 4.2.1. [BCS00] Given a Finsler manifold (M, F ). There exists
an unique linear connection called the Chern connection ∇ which satisfy the
following conditions:
a) d(dxi ) − dxj ∧ ωji = −dxj ∧ ωji = 0 (Torsion freeness)
b) dgij − gkj ωik − gik ωjk = 2Aijs δyF (Almost g-compatibility)
s
Proof. The proof of this theorem is omitted but can be found in either of [BCS00]
and [Mat86].
4.3 Geodesics
As for geodesics in the Riemannian geometry, a similar approach in Finsler geometry for geodesics is conducted to create an equation governing whether a
curve is a geodesic or not and as such serving as an equivalent denition of a
66
4.3 Geodesics
4 FINSLER GEOMETRY
geodesic curve in the Finsler geometry.
The denition of the length of a curve in the Finsler geometry follows the same
concept as in the Riemannian geometry. This means using the Finsler structure
for "weighting" the dierent regions of the manifold. One big dierence between
the two geometries is that the length of a curve in the Finsler geometry is not
independent of the direction in which the path is traveled as is the case in the
Riemannian geometry.
Denition 4.3.1 (Length of curve). Given a Finsler manifold (M, F ) and a
continuously dierentiable curve γ : I[a,b] → M on M . The length of the curve,
denoted l(γ), is given by:
Zb
l(γ) =
F (γ(t), γ 0 (t))dt.
(4.3.1)
a
To dene the concept of geodesics in the Finsler geometry, the rst variation of
arc length is used. It is the derivative of the arc length of curves γ(t, u) which
will be explained below.
Given a piecewise smooth curve γ(t), 0 ≤ t ≤ r with a partition:
0 = t1 < t2 < . . . < tk = r
(4.3.2)
such that γ is smooth on every part of the partition. Then the length of γ (arc
length) is given by:
l(γ) =
k Zti
X
i=2t
F (γ(t), γ 0 (t))dt.
(4.3.3)
i−i
Now consider the rectangular domain:
= {(t, u) : 0 ≤ t ≤ r, − < u < }
(4.3.4)
A piecewise smooth variation of γ(t) is a continuous map γ(t, u) from to the
manifold M which satises that on each rectangle:
I[ti−1 ,ti ] × I(−,)
(4.3.5)
the map is smooth and that γ(t, 0) = γ(t).
For every xed value of u the mapping γ(t, u) give rise to a t-curve and similarly
if t is constant it is an u-curve . The length of a t-curve (denoted l(u)) for a
given variation γ(t, u) is given by:
l(u) =
k Zti
X
i=2t
∂γ
(t, u) dt.
F γ(t, u),
∂t
i−1
67
(4.3.6)
4.3 Geodesics
4 FINSLER GEOMETRY
Now dierentiating (4.3.6) with respect to u gives the rst variation of arc
length . To see how this is done and to get a more thorough explanation of this
subject see 5.1 in [BCS00].
Then an alternative equivalent denition to the general Denition 2.3.4 given
in Chapter 2 for a curve being geodesic in the Finsler geometry is given below.
Denition 4.3.2 (Geodesics in Finsler geometry). Given a Finsler manifold
(M, F ) and a piecewise smooth curve γ : I[0,r] → M on M . Then γ is a geodesic
if:
l0 (0) = 0
(4.3.7)
for every piecewise smooth variation of γ with xed endpoints.
Proof. The equality of the general denition of geodesics and the specic denition in the Finsler geometry can be found in [BCS00]
For some examples of geodesics in the Finsler geometry and further discussions
of this topic see [BCS00].
In the Riemannian geometry the metric in Denition 3.4.2 gives a length space
for a Riemannian manifold. Hopefully using this "metric" on Finsler manifolds
will also give a length space for Finsler manifolds but unfortunately this is not
the case due to the resulting "metric" is in fact not a metric.
Denition 4.3.3 ("Metric" on Finsler manifold). Let the function d on the
Finsler manifold (M, g) be given by:
d(p1 , p2 ) = inf l(Φ)
(4.3.8)
where Φ is the set of continuous curves of the form γ : I[a,b] → M,
γ(a) = p1 and γ(b) = p2 .
The function d dened in Denition 4.3.3 satises the rst and third condition
in Denition A.0.9 for being a metric but not the second property of being a
metric, namely:
d(x1 , x2 ) = d(x2 , x1 ).
(4.3.9)
Example 4.3.1 (A Finsler manifold for which d is not a metric). Finsler manifolds whose Finsler structure is not absolute homogeneous:
F (x, λy) = |λ|F (x, y)
for every λ ∈ R
(4.3.10)
does not necessarily satisfy the symmetry condition for d being a metric. This
68
4.3 Geodesics
4 FINSLER GEOMETRY
can be intuitively realized by observing that when moving in the opposite direction of the path dening d(x1 , x2 ), i.e the inmum length of paths between x2
and x1 , in the point x, the value of the Finsler structure does not necessarily
coincide with the value of the Finsler structure at x when moving between x1
and x2 and hence the lengths of the two paths are not always equal.
The Finsler manifold (R2 , F ) seen in Example 4.1.2 and dened as:
F (x, y) =
rq
y14 + y24 + µ(y12 + y22 )
(4.3.11)
where y = [y1 , y2 ]. This Finsler manifold is absolute homogeneous and hence d
is a metric on (R2 , F ).
Remark 4.3.1. A function which satisfy the rst and third condition of being a
metric is called a semi-metric .
To show that d is a semi-metric the following lemma is used.
Lemma 4.3.1. [BCS00] Given a the Finsler manifold (M, F ). There exists
at every point x ∈ M a local coordinate system φ : Ū → Rn with the following
properties for an open set U where Ū is the closure of U and a constant c > 1:
a) Ū is compact, φ(x) = 0 and φ maps U dieomorphically on an open ball in
Rn .
b) There exists a constant c > 1 such that:
1
|y| ≤ F (x, y) ≤ c|y|
c
and F (x, −y) ≤ c2 F (x, y)
(4.3.12)
for every y = yi ∂x∂ i ∈ Tp M and p ∈ Ū . Furthermore here, |y| = δij yi yj
where
(
0, if x 6= y
δij =
(4.3.13)
1, if x = y
p
which is known as the Kronecker delta.
c) Given x1 , x2 ∈ U . Then:
1
|φ(x2 ) − φ(x1 )| ≤ d(x1 , x2 ) ≤ c|φ(x2 ) − φ(x1 )|.
c
(4.3.14)
d) Given x1 , x2 ∈ U . Then:
1
d(x2 , x1 ) ≤ d(x1 , x2 ) ≤ c2 d(x2 , x1 ).
c2
69
(4.3.15)
4.3 Geodesics
4 FINSLER GEOMETRY
Proof. a) Choose a local coordinate system φ : W → Rn where W ⊂ M and
φ(x) = 0. Let B(r) be an open ball dened by:
n
p o
B(r) = xi ∈ Rn : xi < δij
Take a r > 0 such that:
Then the inverse image:
(4.3.16)
ˆ ⊂ φ(W )
B(r)
(4.3.17)
φ−1 (B(r))
(4.3.18)
is precompact and hence φ : W → Rn satisfy the rst condition of the lemma.
The rest of the conditions are omitted here but can be found in [Mat86].
Before proving the rst and third properties for being a metric for d, the "topology generated by d" will be proved to equal the topology of the manifold.
The balls which will generate the topology are the forward metric balls which
are dened below.
Denition 4.3.4 (Forward metric ball). Given a Finsler manifold (M, F ).
forward metric ball B + (x, r) centered at x with radius r is given by:
B + (x, r) = {p ∈ M : d(x, p) < r} .
A
(4.3.19)
Remark 4.3.2. Note that the order of x and p is important in the denition of
forward metric ball due to the non-symmetry of d.
Example 4.3.2 (Forward metric ball). Consider the Finsler structure dened
by the Riemann metric in Example 3.2.3 which is is just the Euclidean inner
product. Then the forward metric balls in the Finsler manifold (Rn ), F ) are just
the open balls dened by the Euclidean metric.
Theorem 4.3.1. [BCS00] The topology of the Finsler geometry coincides with
the topology generated by the forward balls when seen as a metric space, i.e a
set is open if around every point in the set there exists a forward ball containing
the point and being contained by the set.
Proof. The proof of this theorem is omitted here but can be found in [BCS00].
Now it is time to prove that d satises the rst and third condition of a metric.
Theorem 4.3.2. [BCS00] Given a Finsler manifold (M, F ). Then the function
d dened in Denition 4.3.3 satises for every x1 , x2 ∈ M :
70
4.3 Geodesics
4 FINSLER GEOMETRY
a) d(x1 , x2 ) = 0 ⇐⇒ x1 = x2 .
b) d(x1 , x3 ) ≤ d(x1 , x2 ) + d(x2 , x3 ).
Proof. The proof of this theorem is omitted here but can be found in [BCS00].
Because d is not a metric, a Finsler metric is not necessarily a length space and
hence the Hopf-Rinow-Cohn-Vossen Theorem 2.3.3 is not readily accessible. Instead, a variant of this theorem will be developed and to do that, the Finsler
geometry counter part to Cauchy sequences will be needed.
Denition 4.3.5 (Forward Cauchy sequence). Given a Finsler manifold (M, F ).
A sequence {xi } ∈ M is a forward Cauchy sequence if, for every > 0, there
exists an N ∈ N such that:
N ≤ i < j =⇒ d(xi , xj ) < .
(4.3.20)
The reason Forward Cauchy sequences are needed are for the counterpart of
Hopf-Rinow-Cohn-Vossen theorem in the Finsler geometry which naturally should
state something concerning complete spaces but due to the non-symmetry of the
"metric" function, Cauchy sequences are not dened and hence Forward Cauchy
sequence will be used to dene "complete" spaces in Finsler geometry.
Denition 4.3.6 (Forward complete space).
Given a Finsler manifold (M, F ).
The Finsler manifold is a forward complete space if every Forward Cauchy sequence converges in M .
The last thing needed to dene is the Finslerian counterpart to boundedly compact spaces which is not surprisingly given by:
Denition 4.3.7
(Forward boundedly compact space). Given a Finsler manifold (M, F ). A subset of the Finsler manifold is forward bounded if it is contained
in a forward metric ball. The Finsler manifold is a Forward boundedly compact
space if every closed and forward bounded subset is closed.
Finally we can state and prove the Hopf-Rinow-Cohn-Vossen theorem in the
Finsler geometry.
Theorem 4.3.3. [BCS00] Given a connected Finsler manifold (M, F ). Then
the following statements are equivalent:
a) The space (M, F ) is forward complete.
b) The space (M, F ) is a forwardly bounded space.
71
4.3 Geodesics
4 FINSLER GEOMETRY
and both these conditions gives that there exists a shortest path γ between every
point x1 , x2 ∈ M .
Proof. The proof of this theorem is omitted here but can be found in [BCS00].
72
A METRIC SPACES
Appendices
The following appendices are meant to cover the important results and denition which are not directly related to the subjects of this thesis but still needed
in order to fully understand all the material covered in this thesis. One goal of
these appendices is to be self-contained so no outside resources are needed for
the reader to understand the material but still at the beginning of each appendices there will be a part devoted to mentioning other sources which gives more
detailed exposition of the subjects covered in the respective appendices.
A
Metric spaces
Metric spaces are spaces for which a metric (function which gives a distance for
every pair of points) is dened. The concept of metric spaces is important in this
thesis both from a topological point of view because the metric gives a topology
and hence the open sets of the metric spaces and because this thesis will mostly
focus on length spaces which are metric spaces with some restrictions. Most of
the material in this appendix can be found in either of [Mor05] or [Rud76].
Denition A.0.8 (Real coordinate space).
The real
set of all n-tuples of R, for any positive integer n.
coordinate space, Rn is the
Example A.0.3 (Real coordinate space). An example of an element x ∈ R3 is
x = [0, 12 , π]
A metric is a function which gives a distance (non-negative number) for every
pair of elements in the set.
Denition A.0.9 (Metric).
A metric on a set X is a function d : X × X → R
that satises for all x, y, z ∈ X the following:
a) d(x, y) = 0 ⇐⇒ x = y
b) d(x, y) = d(y, x) (Symmetric)
c) d(x, y) ≤ d(x, z) + d(z, y) (Triangle inequality)
Remark A.0.3. Often in the literature a fourth condition is required for a func-
tion to be a metric, d(x, y) ≥ 0. This denition is equivalent with the one given
below due to the other three statements imply the additional condition which
is shown in Proposition A.0.1.
Example A.0.4 (Metric). The
discrete metric:
(
0, if x = y
d(x, y) =
1, if x 6= y
(A.0.21)
satises a) − c) on R. Here a) − c) follows trivially from the Denition A.0.9 of
metric and the denition of the discrete metric.
73
A METRIC SPACES
Remark A.0.4. The discrete metric will be used extensively throughout this
thesis in dierent examples because using this metric often give non-intuitive
examples which let the reader get a better picture of what the specic object in
the example could look like.
Example A.0.5 (Metric). The function:
d(x, y) = max(x, y)
(A.0.22)
on R+ is not a metric. Here b) and c) follows trivially from the Denition A.0.9
of metric but a) is not fullled.
Example A.0.6 (Metric). The function:
d(x, y) = min(x, y)
(A.0.23)
on R is not a metric. Here a) and c) are not fullled. For example let x = 1
and y = 0. Then d(x, y) = 0 and x 6= y. Therefore a) is not fullled. Let
x = 1, y = 1 and z = 0. Then d(x, y) = 1 and
d(x, z) + d(z, y) = 0,
(A.0.24)
so the triangle inequality is not fullled.
Proposition A.0.1. A metric d on a set X satises the following
for all x, y ∈ X :
d(x, y) ≥ 0.
(A.0.25)
Proof. Consider:
2d(x, y) = d(x, y) + d(x, y) =
= d(x, y) + d(y, x) ≥ (by the triangle inequality)
≥ d(x, x) = 0
Because of this:
d(x, y) ≥ 0.
(A.0.26)
A metric space is simply a space with a metric dened on the set. Almost every
space considered in this thesis will be metric spaces and hence these are very
important.
Denition A.0.10
(Metric space). A
where X is a set and d is metric on X .
metric space is an ordered pair (X, d)
74
A METRIC SPACES
Example A.0.7 (Metric space). The ordered pair (R, d), where d is the discrete
metric is a metric space.
Denition A.0.11 (Ball).
Given a metric space (X, d). The open
at p ∈ X with radius r and metric d:
ball centered
Bd (p, r) := {x ∈ X : d(x, p) < r} .
A
(A.0.27)
closed ball centered at p ∈ X with radius r and metric d:
Bd (p, r) := {x ∈ X : d(x, p) ≤ r} .
(A.0.28)
Example A.0.8 (Ball). Given the metric space (Z, d) where d is the discrete
metric from Example A.0.4. The open ball Bd (0, 1) equals the set {0} and the
closed ball Bd (0, 1) equals the set Z.
A set in a metric space is bounded if there is a ball which contains the set. In
Chapter 2 the concept of boundedly compact will be important and as the name
suggest, such sets are among other things bounded.
Denition A.0.12 (Bounded).
Given a metric space (X, d). A subset Y of X
is bounded if for some x ∈ X there exists a ball Bd (x, r) of nite radius which
contains Y . The space (X, d) is called a bounded metric space if X is bounded.
Example A.0.9 (Bounded). Given the metric space (R, d), where
d(x, y) = |x − y|.
(A.0.29)
For example the intervals I[−10π,100] , I(−3,3) and I[e, 27 ] are contained by Bd (0, 400),
so these sets are bounded. The interval I(0,∞) is not bounded.
Example A.0.10 (Bounded). Given the metric space (R, d), where d is the
discrete metric. This space is bounded which can be seen by for example
Bd (x, r) ⊂ R for every x ∈ R where r > 1.
In a metric space, the metric induces the open sets. A set is open if there is for
every point a ball centered at the point contained in the set.
Denition A.0.13 (Open set in a metric space). Given a metric space (X, d).
A subset U of X is open if for every x ∈ U , there exists an open ball Bd with
center at x such that B ⊂ U .
75
A METRIC SPACES
Example A.0.11 (Open set). Given the metric space (R, d),
where
d(x, y) = |x − y|.
(A.0.30)
U = {x ∈ R : d(0, x) < 1} .
(A.0.31)
Choose the subset
This subset U is an open set in the metric space (R, d).
Denition A.0.14
(Sequence). Given a set X . A sequence is an ordered list
of elements x0 , x1 , x2 , . . . , xk ∈ X which will be denoted {xk }. A sequence is
called nite if the number of elements in the sequence is nite otherwise it is
innite.
Example A.0.12 (Sequence). Given the set R. The following list:
x0 = 1, x1 =
1
1
, . . . , xk =
,...
2
k+1
(A.0.32)
is a sequence.
Remark A.0.5. Henceforth a sequence will be assumed to be innite if not otherwise stated.
The notion of Cauchy sequences is a corner stone in analysis and they are used
extensively both in general and in this thesis. Informally a Cauchy sequence is
a sequence for which the distance between elements becomes arbitrarily small
for later elements of the sequence. Formally it is dened as:
Denition A.0.15
(Cauchy sequence). Given a metric space (X, d). A sequence x1 , x2 , x3 , . . . ∈ X is a Cauchy sequence if for every > 0 there exists an
N ∈ N such that for every n, m > N ,
d(xn , xm ) ≤ .
(A.0.33)
Example A.0.13 (Cauchy sequence). Given the metric space (R, d), where
d(x, y) = |x − y|.
(A.0.34)
x0 = 0, x1 = 0.9, x2 = 0.99, x3 = 0.999, . . .
(A.0.35)
The sequence
is a Cauchy sequence.
76
A METRIC SPACES
Example A.0.14 (Cauchy sequence). Given the metric space
is the discrete metric. The sequence
x0 = 0, x1 = 0.9, x2 = 0.99, x3 = 0.999, . . .
(R, d),
where d
(A.0.36)
is not a Cauchy sequence. This can be seen by choosing an 0 < < 1. Then
there is not an N ∈ N such that whenever n, m > N and n 6= m,
d(xn , xm ) ≤ .
(A.0.37)
This is because d(xn , xm ) = 1 and hence {xk } is not a Cauchy sequence.
When Cauchy sequences are dened, complete spaces has the dening property
that every Cauchy sequence converges to a limit in the space.
Denition A.0.16 (Complete). Given a metric space (X, d). The space X
complete if every Cauchy sequence has a limit which belongs to X .
is
Example A.0.15 (Complete). The metric space (Q, d), where d(x, y) = |x−y|
is not a complete space. For example the Cauchy sequence
3, 3.1, 3.14, 3.141, 3.1415. . . .
(A.0.38)
has the limit π but π ∈/ Q.
Remark A.0.6. If all the accumulation points of Q are added to Q, this new set
will be the completion of Q and in this case it is R.
Example A.0.16 (Complete). The metric space (R, d), where d(x, y) = |x − y|
is a complete space.
Denition A.0.17
(Lipschitz continuous). Given two metric spaces (X, dx )
and (Y, dy ). A function f : X → Y is Lipschitz continuous if there exists a real
C ≥ 0 such that for all x1 , x2 ∈ X :
dx (f (x1 ), f (x2 )) ≤ C · dy (x1 , x2 )
(A.0.39)
Remark A.0.7. When a function is Lipschitz continuous it can intuitively be
thought of as when the absolute value of slope of the straight line joining the
function values is bounded by a real number. This can be seen by dividing
A.0.39 by dy (x1 , x2 ) for x1 , x2 ∈ X , y1 , y2 ∈ Y and this gives:
dx (f (x1 ), f (x2 ))
≤ C.
dy (x1 , x2 )
77
(A.0.40)
A METRIC SPACES
Remark A.0.8. A real C satisfying Denition A.0.17 is a Lipschitz constant of
f , a minimal constant is called the dilation of f , while a map with constant 1
is called non-expanding .
One classical case where Lipschitz continuity is used is as a condition in the
Picard-Lindelöf theorem, which states when a solution to an ordinary dierential equation exists and is unique for an initial-value problem (see [NSS11]). In
this thesis Lipschitz continuous will be useful together with the following theorem and Remark A.0.9 which says that if a function is shown to be Lipschitz
continuous then it is also continuous.
Theorem A.0.4. Given metric spaces (X, dx ) and (Y, dy ) with a Lipschitz continuous function f : X → Y . Then f is uniformly continuous on X .
Remark A.0.9. Using that uniform continuity implies continuity gives that Theorem A.0.4 implies continuity.
Proof. The proof of this theorem is omitted but can be found in [MVJ10].
78
B TOPOLOGICAL SPACES AND TOPOLOGY
B
Topological spaces and topology
The topic of topology deals essentially with subjects such as continuity, open
sets, connectedness and compactness. These terms are generally known for students taking a couple of calculus courses but in topology they are generalized
from subsets of R to arbitrary sets. This thesis rely heavily on topological result
and in particular Chapter 2 uses the results from this appendix extensively. In
particular, compactness will be very important for this thesis when studying
when shortest paths exist for dierent length spaces. Perhaps the most important result proven in this appendix is the equivalence of what is known as
sequential compactness and compactness for metric spaces. This result will later
on be applied many times in several dierent proofs. Most of the results in this
appendix can be found in the text books [Rud76] and [Mor05] where especially
the former give a thorough exposition of the basics of this subject.
B.1 Topological spaces
A topological space is a space which characterizes the topological properties
such as continuity, open sets, connectedness and compactness. If a space is not
topological, then crucial concepts such as continuity, open sets, etc. are not
dened which are not optimal due these concepts are corner stones in analysis.
Because of that the spaces in this thesis are topological. Furthermore in this
section several important topological concepts will be dened with corresponding examples given.
Denition B.1.1
(Closed). A set X is
x, y ∈ X , x • y ∈ X .
closed under an operation (•) if for all
Example B.1.1 (Closed). Given the set Z. This set is closed under the operations addition (+), subtraction (−) and multiplication (·).
Example B.1.2 (Closed). Given the set Z. This set is not closed under the
operation division (/). For example for 3, 4 ∈ Z, but 43 ∈/ Z.
Denition B.1.2 (Topological space). The set X
subsets of X , denoted σ , is a topological space if:
together with a collection of
a) ∅ and X belongs to σ
b) σ is closed under union
c) σ is closed under nite intersection
Example B.1.3 (Topological space). Given the set:
X = {−3, −2, −1, 0, 1, 2, 3}
(B.1.1)
and the collection of subsets σ = {∅, X}. Then (X, σ) is a topological space.
79
B.1 Topological spaces
B TOPOLOGICAL SPACES AND TOPOLOGY
Example B.1.4 (Topological space). Given the set X = {−1, 0, 1} and collection of subsets σ = {∅, X, {1}}. Then (X, σ) is also a topological space.
Given a topological space, the topology is then the collection σ of subsets.
Denition B.1.3
(Topology). A
σ satisfying Denition B.1.2.
Example B.1.5
subsets:
(Topology).
topology to a set X is a collection of subsets
Given the set X = {−1, 0, 1}. The collection of
(B.1.2)
σ = {∅, X, {−1} , {0} , {−1, 0}}
is a topology.
Example B.1.6
subsets:
(Topology).
Given the set X = {−1, 0, 1}. The collection of
(B.1.3)
σ = {∅, X, {−1} , {0}}
is not a topology on the set X since the subset {−1, 0} (which is the union
does not belong to σ.
{−1} ∪ {0})
Now two dierent topologies will be dened. The discrete topology is denable
for every set and the standard topology is denable for every metric spaces.
Denition B.1.4 (Discrete topology).
the collection of every subset of X .
Example B.1.7
lection of subsets
(Discrete topology).
Given a set X . The
discrete topology is
Given the set X = {−1, 0, 1}. The col-
σ = {∅, X, {−1} , {0} , {1} , {−1, 0} , {−1, 1} , {0, 1}}
(B.1.4)
is the discrete topology of X .
Now the open set of the topological spaces are the members of the topology.
Denition B.1.5 (Open set in topological spaces). Given a topological space
X with topology σ . The open sets are the members of σ .
Example B.1.8 (Open set). Given the topology:
σ = {∅, {−1, 0, 1, } , {−1} , {0} , {−1, 0}}
(B.1.5)
of the topological space (X = {−1, 0, 1, } , σ). Then for example {−1, 0} is an
open set in X .
80
B.1 Topological spaces
B TOPOLOGICAL SPACES AND TOPOLOGY
The usual topology considered in metric spaces is the standard topology (natural topology) which uses the metric to dene the open set in the following way:
Denition B.1.6 (Standard topology). Given a topological space (X, σ) with
a metric d. Then σ is the standard topology if the following holds:
A subset of X is an open set ⇐⇒ Around every point in the set there exists an
open ball contained in the set
Example B.1.9 (Standard topology). Considering the topological space (R, σ)
with the standard topology induced by the Euclidean metric d(x, y) = |x − y| in
Example F.0.19. For example the following sets are open in (R, σ):
{x ∈ R : 0 < x < 1}, x ∈ R : −π < x <
4
229
If not otherwise specied a topological space with the standard topology has
the Euclidean metric as its metric in this appendix.
The concept of connectedness in a topological space is not to be confused
with another similar concept of path-connectedness. Shortly, a space is pathconnected if there exists a path connecting every pair of points in the space.
Connectedness, which will be dened below does not imply path-connectedness
but path-connectedness instead implies connectedness.
Denition B.1.7 (Connected).
is disconnected if it is the union
connected.
Given a topological space (X, σ). This space
of two disjoint open sets, otherwise (X, σ) is
Example B.1.10 (Connected). Considering the topological space (R, σ) with
the standard topology. This space is connected because there are no two intervals
which are both open and is a covering of R. In contrast, if the intervals need
not be open we can cover R by the intervals:
{x ∈ R : ∞ < x ≤ 0}
(B.1.6)
{x ∈ R : 0 ≤ x < ∞}
(B.1.7)
and
which is a covering of R but they are not disjoint open sets.
Example B.1.11 (Connected). Considering the topological space (Z, σ) with
the discrete topology. This space is disconnected. For example the open sets {0}
and Z \ {0} cover Z and are open sets because every set in the discrete topology
is open.
Generalizing continuous functions to functions between topological spaces are
formally dened below. Intuitively, this denition can be understood by thinking of points belonging to the same open set as being "close together" and thus
this denition will state something similar to the epsilon-delta denition.
81
B.1 Topological spaces
B TOPOLOGICAL SPACES AND TOPOLOGY
Denition B.1.8 (Continuous in topological spaces). Given the topological
spaces (X, σ1 ) and (Y, σ2 ). A function f : X → Y is continuous (continuous
map) if for all V ∈ σ2 , the inverse image:
f −1 (V ) = {x ∈ X|f (x) ∈ V }
(B.1.8)
is a member of σ1 (see Figure 11).
Figure 11: A continuous function
Example B.1.12 (Continuous). Given the topological space (X, σ), where X =
{−1, 0, 1, } and σ is the trivial topology σ = {∅, X}. The function:
f : (X, σ) → (X, σ), f (x) = x.
(B.1.9)
is a continuous function.
For the open set X , the inverse image f −1 (X) = X which is an open set so f
is continuous.
Example B.1.13 (Continuous). Given the topological spaces (X, σ1 ), (Y, σ2 ),
where X = {−1, 0, 1, }, Y = {0, 1, 2, }, σ1 is the trivial topology σ = {∅, X} and
σ2 = {∅, 2, Y }. The function:
f : (X, σ1 ) → (Y, σ2 ), f (x) = x + 1.
is not continuous because f −1 (2) = 1 ∈/ σ1 .
82
(B.1.10)
B.1 Topological spaces
B TOPOLOGICAL SPACES AND TOPOLOGY
For the open set X , the inverse image f −1 (X) = {−1, 0} is not an open set so
f is not continuous.
Another key concept in topology are neighborhoods. A neighborhood to a point
is a set containing an open set and for which the point belongs to the open set.
Denition B.1.9 (Neighborhood). Given a topological space X and a point
x ∈ X . A neighborhood of x is a set V that contains an open set U such that:
x ∈ U ⊆ V.
(B.1.11)
Further on the notation (X, σ) for a topological space will mostly be simplied
to X and the standard topology will be assumed in this appendix together with
the Euclidean metric unless otherwise stated.
Example B.1.14 (Neighborhood). Let the topological space be given by:
(X = {−1, 0, 1, } , σ)
(B.1.12)
where σ is given in Example B.1.8. A possible neighborhood of 0 is the set
V = {−1, 0}. In Example B.1.8 we see that V = {−1, 0} is an open set and
V ⊆ V . So V is a neighborhood of 0 in the topological space (X, σ).
One further important topological concept are accumulation points. Accumulation points are points in a set which can be "approximated" by other points
in the set. This is formalized as followed.
Denition B.1.10
(Accumulation point). Given a topological space X and a
subset Y ⊂ X . A point x ∈ X is an accumulation point (limit point) of Y if
every open set containing x in X contains at least one point in Y which is not x.
Example B.1.15 (Accumulation point). Given the topological space
the standard topology and the subset
Y = [x, y] ∈ R2 : x2 + y 2 < 1 \ {0}.
R2
with
(B.1.13)
The set of accumulation points to Y is
[x, y] ∈ R2 : x2 + y 2 ≤ 1 .
Example B.1.16 (Accumulation point). Given the topological space
the discrete topology and the subset
Y = [x, y] ∈ R2 : x2 + y 2 < 1 \ {0}.
83
(B.1.14)
R2
with
(B.1.15)
B.1 Topological spaces
B TOPOLOGICAL SPACES AND TOPOLOGY
The set of accumulation points to Y is ∅. This is because at every x ∈ Y , the
open set x do not contain any other points in Y .
When accumulation points are dened, the notion of closed set is easy to dene.
Denition B.1.11 (Closed set).
tion points.
A set is
closed if it contains all its accumula-
Remark B.1.1. Another used denition for closed sets is that a set is closed if it
is the complement of an open set. That these denitions are in fact equivalent
for the special case of the standard topology is shown in Lemma B.2.1.
Example B.1.17 (Closed set). Given the topological space
dard topology and the subset
Y = [x, y] ∈ R2 : x2 + y 2 < 1 \ {0}.
R2
with the stan(B.1.16)
Here Y is not a closed set because as seen in Example B.1.15 the accumulation
points
[x, y] ∈ R2 : x2 + y 2 = 1 ∪ {0}
(B.1.17)
is not contained in Y but instead the set
[x, y] ∈ R2 : x2 + y 2 ≤ 1
(B.1.18)
is a closed set.
Example B.1.18 (Closed set). Given the topological space R2 with the discrete
topology and the subset
Y = [x, y] ∈ R2 : x2 + y 2 < 1 \ {0}.
(B.1.19)
Here Y is a closed set because as seen in Example B.1.16 the set does not have
any accumulation points and ∅ ⊂ Y . In fact every set is both open and closed
with the discrete topology. This property is called a clopen set.
The notion of limits in topological spaces is readily generalized from the epsilondelta denition as follows:
Denition B.1.12 (Limit). A limit of
X is an element a ∈ X which satises:
a sequence {xk } in a topological space
The element a ∈ X is a limit of {xk } ⇐⇒ For every neighborhood (see Denition B.1.9) U of a there is an N ∈ N such that xn ∈ U for every n ≥ N .
The sequence {xk } is called
convergent if a limit exists otherwise divergent.
84
B.1 Topological spaces
B TOPOLOGICAL SPACES AND TOPOLOGY
Remark B.1.2. This is equivalent in R with: A limit of a sequence {xk } in R is
an unique element a ∈ X which satises:
The element a ∈ X is a limit of {xk } ⇐⇒ For every > 0 there is an N ∈ R+
such that:
|xn − a| < for every n > N.
(B.1.20)
A sequence {xk } in Rn has a limit if every coordinate in {xk } separately has a
limit.
Remark B.1.3. The limit of a sequence will be denoted by:
lim xk
k→∞
(B.1.21)
Example B.1.19 (Limit). Given the sequence {xk } in R, where xk = n1 . This
sequence has the limit 0. Choose an and let N = 1 . Now for every n > N ,
|xn − 0| < .
Remark B.1.4. A sequence {xk } in Rn is monotone if it is increasing or decreasing in every coordinate of {xk }.
The following theorem known as the monotone convergence theorem gives that
for monotone sequences, the existence of a limit and being bounded are equivalent.
Theorem B.1.1 (Monotone convergence Theorem). [Mor05] Let
monotone sequence in Rn . The following statements are equivalent.
{xk }
be a
The sequence {xk } has a limit in Rn ⇐⇒ {xk } is bounded
Proof. Assume the sequence {xk } is in R and monotone increasing. Because
{xk } is bounded, the supremum a = sup xk is nite. Now choose an , then
there exists a xN such that:
k
xN > a − (B.1.22)
because otherwise a would not be a supremum.
Now utilizing that {xk } is increasing gives for all n > N ,
|xn − a| = a − xn ≤ a − xN < ,
(B.1.23)
where the last inequality is derived from (B.1.22). But this is the denition
that a is the limit of {xk }. Exactly the same reasoning can be used to show
85
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
that if {xk } is decreasing, it has a limit, a = inf xk . This gives that a bounded
k
sequence in R has a limit.
If {xk } is a sequence in Rn the same procedure as above can be used for every
coordinate of {xk }, to show that it has a limit.
B.2 Compactness
In this section a number of important result concerning compact spaces will be
given and to successfully accomplish this some further topological concepts are
needed. Perhaps the most important result in this section is the equivalence of
compactness and sequential compactness (which will be dened later) in metric
spaces.
The compactness of a space is another property of the topology. This notion is
very abstract and hard to be put into word for a general topological space. In
a metric space, compactness can be thought of as when choosing points from
the set, eventually two of the point needs to be arbitrarily close together. The
abstract denition of compactness is given below.
Denition B.2.1 (Compact). Given a topological space X . The space
compact if for every collection of open subsets of X , {Ui }i∈A, such that:
X⊂
[
Ui ,
X is
(B.2.1)
i∈A
there is a nite subcovering, i.e there is a nite B ⊆ A such that:
[
X⊂
Ui .
(B.2.2)
i∈B
Sometimes this denition is hard to work with and due to this the main result
in this section is the equivalence of sequential compactness and compactness in
metric spaces. In some cases in this thesis, sequential compactness will be easier
to work with in some proofs, specically in Chapter 2.
Before the equivalence of the two types of compactness is proven, another result
for subsets of E n called the Heine-Borel Theorem B.2.1 will be proven. This
states that a subset X of E n is compact if and only if X is closed and bounded.
To prove this theorem, interior points will be used. An interior point to a subset
is a point such that, there exists a neighborhood around the point contained in
the subset.
Denition B.2.2 (Interior point). Given a topological space X with metric d.
A point x is an interior point of a subset Y ⊆ X if there exists an open ball
Bd (x, r) such that:
Bd (x, r) ⊂ Y.
86
(B.2.3)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Remark B.2.1. Note that with the standard topology, every point in an open
set is an interior point.
Example B.2.1 (Interior point). Given the topological space R2 with the discrete metric. The set R2 has each of its points as an interior point which can
be seen by choosing x ∈ R2 and the open ball:
(B.2.4)
Bd (x, 1) ⊂ R2 .
Example B.2.2 (Interior point). Given the topological space
crete metric. The set:
R2
with the dis(B.2.5)
X = [x, y] ∈ R2 : x2 + y 2 ≤ 1
has each of its points as an interior point which can be seen by choosing x ∈ X
and the open ball:
(B.2.6)
Bd ([x, y], 1) ⊂ X.
Example B.2.3 (Interior point). Given the topological space
clidean metric. The set:
R2
X = [x, y] ∈ R2 : x2 + y 2 ≤ 1 = Bd (0, 1)
with the Eu(B.2.7)
have points which are not interior points. More precisely, every point such that:
x2 + y 2 = 1
(B.2.8)
is not an interior point of X because there does not exists an open ball containing such a point and such that the open ball is contained in X .
The following three lemmas will be used to prove the Heine-Borel Theorem
B.2.1. The rst of these lemmas concerns when a complement to an open set is
a closed set.
Lemma B.2.1. [Rud76] Given a topological space X with the standard topology. Then the following statements are equivalent:
A subset Y ⊂ X is open ⇐⇒ The complement Y c is a closed set.
Proof. =⇒: Assume Y is open.
Choose an accumulation point y ∈ Y c . Then every open ball around y contains
a point x ∈ Y c . This gives that y is not an interior point of Y and since Y is
an open set, y ∈ Y c .
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B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
⇐=: Assume Y c is closed.
Choose y ∈ Y . Then y ∈
/ Y c and y is not an accumulation point of Y c . Thus
there exists an open ball B around y such that:
B ∩ Y c = ∅.
(B.2.9)
Then B ⊂ Y and thus y is an interior point of Y .
The following lemma is not only essential to prove the Heine-Borel theorem but
will also be used repeatedly in Chapter 2.
Lemma B.2.2. [Rud76] Given a topological space with the standard topology.
Then a closed subset X of a compact set Y is compact.
Proof. Consider the collection BX of open set which covers X . Now the set
U = X c is an open set and the collection:
BY ∪ U
(B.2.10)
is an open covering of Y . Because Y is compact there exists a nite subcovering
of BY ∪ U ,
B0Y ∪ U
(B.2.11)
which covers Y . The sets X and U are disjoint by construction which makes
the collection:
B0X = B0Y \ U
(B.2.12)
a nite subcovering of X .
The last lemma is known as the Cantor's intersection lemma. This is a technical
result which states, a sequence of closed and bounded subsets of E n which are
"nested" has a non-empty intersection.
Lemma B.2.3 (Cantor's intersection lemma). [Rud76] Given a sequence {Xk }
of closed and bounded subsets of E n such that:
X0 ⊇ X1 ⊇ · · · Xk ⊇ Xk+1 · · ·
This gives the following:
\
Xk 6= ∅
k
88
(B.2.13)
(B.2.14)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Proof. Consider the sequence {ak }, where ak is the inmum of Xk at each co-
ordinate and since Xk is closed ak ∈ Xk . This sequence will be monotonic
increasing, i.e will be increasing for every coordinate separately and since it is
bounded, it must have a limit a from Theorem B.1.1.
Now choose any i ≥ 0. The subsequence {ak }k≥i is contained in Xk and converges to a. Since Xk is closed, a ∈ Xk . This is true for every Xk so:
\
a∈
Uk
(B.2.15)
k
Now the ground work to prove the Heine-Borel theorem is ready and what follows is the theorem with a corresponding proof.
Theorem B.2.1 (Heine-Borel theorem). [Rud76] Given a subset
The following statements are equivalent:
X
X
of E n .
is compact ⇐⇒ X is closed and bounded
Proof. =⇒: Assume X is compact.
First observe the following: Let a be an accumulation point in X . Then any
nite collection C of open sets, where the open sets U ∈ C are such that there
exist neighborhoods VU of a. These open sets are not an open covering which
can be seen by considering the intersection of the neighborhoods VU of a. This
set is a neighborhood W of a because every neighborhood contains open set and
the intersection of open sets is open, so W contains an open set and thereby is
a neighborhood. Since a is an accumulation point there has to exists a b ∈ W
which belongs to X . From this it follows that C is not a covering.
Assume X is not closed. Then there exists an accumulation point a such that
a∈
/ X . Now consider the collection D of neighborhoods Ux of x for every x ∈ X
where Ux are open sets, chosen such that they do not intersect a neighborhood
Va of a. This collection D is an open covering of X . Now considering any nite
subcollection of D. This cannot be an open covering of X as seen above with
C. But then X is not compact which is a contradiction. So X is closed.
Consider the collection E of open balls:
{Bd (x, r)}r∈R+ .
(B.2.16)
This is an open covering of X . Now every open subcovering of X must have
a largest ball which contains every other ball in that subcovering. This ball
bounds X . Because of this X is bounded.
⇐=: Assume X is closed and bounded.
Since X is a bounded subset of Rn , a box
Z0 = [−a, a]n
89
(B.2.17)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
which is given by the condition that every dimension of R lies in the interval
I[−a,a] , can always enclose X . From Lemma B.2.2 it is sucient to show that
Z is compact for X to be compact.
Assume that Z0 is not compact. Now there exists an open covering F of Z0
such that there is no nite subcovering of F. Construct 2n boxes by bisecting
the sides of Z0 . These boxes will now have the diameter a.
Because Z0 has not a nite subcovering there exists a box Z1 with an innite
covering from F of the 2n boxes constructed. Now Z1 can be bisected to 2n new
boxes. Repeating this produces a sequence of boxes:
Z0 ⊇ Z1 ⊇ · · · Zk ⊇ Zk+1 · · ·
(B.2.18)
The side length 22ak of the boxes will tend to 0 as k goes to innity. Since the
sequence {Zk } is closed and bounded the Lemma B.2.3 gives that there exists
a x such that:
\
x∈
Zk .
(B.2.19)
k
Since F covers Z0 , there exists an U ∈ F such that x ∈ F. Because U is open,
there exists a ball Bd (x, r) such that:
Bd (x, r) ∈ U.
(B.2.20)
Choosing large enough k , there is a Zk such that:
Zk ⊇ Bd (x, r) ⊇ U.
(B.2.21)
But this gives that the innite subcovering from F of Zk can be replaced by U ,
which yields a contradiction, so Z0 is compact.
Remark B.2.2. Now the Heine-Borel Theorem B.2.1 gives an easy result for
giving examples of compact sets in E n .
Example B.2.4 (Compact). The sets:
and
X = [x, y] ∈ R2 : x2 + y 2 ≤ 1
(B.2.22)
Y = [x, y, z] ∈ R3 : 3x2 − 5y 2 + 2z 2 ≤ 5
(B.2.23)
are compact in E 2 respectively E 3 .
Example B.2.5 (Compact). The set Rn is not compact with the standard topology. This is because the set is not bounded.
90
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Remark B.2.3. Another often used denition of compactness is that a set X is
compact if every sequence in X has a subsequence which converge to a point
in X . This type of compactness is not equivalent with Denition B.2.1 and
is sometimes called sequentially compact . But they are equivalent in E n (and
more generally in metric spaces as seen further on in this appendix) and the
equivalent of the Heine-Borel Theorem B.2.1 for this type of compactness is
called the Bolzano-Weierstrass theorem.
Theorem B.2.2 (Bolzano-Weierstrass theorem). [Mor05] Every bounded sequence in Rn has a convergent subsequence.
Proof. First assume that the sequence {xk } is real and nonnegative. If {xk } has
some negative terms, translate it to a new sequence {yk } such that every term
is nonnegative which is possible due to {xk } is bounded and prove that this
sequence has a convergent subsequence. Then translate back the convergent
subsequence in {yk } to a convergent subsequence in {xk }.
Because the sequence is nonnegative and bounded there exists an integer part
(all integers before the comma) C before the comma sign which occur innitely
many times in the sequence. Let the rst element in the subsequence be xm1 =
C . Now only consider the innitely many elements with the integer part C .
Then there exists a rst decimal place c1 which occurs innitely many times in
the new sequence. Let the second element in the subsequence be xm2 = C.c1
Continue this process to construct a subsequence:
xmk = C.c1 c2 . . . ck . . . .
(B.2.24)
The subsequence {xmk } converges to
a = C.c1 c2 . . .
(B.2.25)
due to choosing > 0. Then there trivially exists an integer N , such that for
every integer i > N ,
d(xmi − a) < .
(B.2.26)
Because xmk will have at least the rst N decimals in common with a.
If {xk } is a sequence in Rn , construct by the above scheme for every coordinate
of xk separately, convergent subsequences:
n
o n
o
n
o
xm1k , xm2k , . . . , xmnk .
(B.2.27)
Then the sequence {Xk }, where
Xk = [xm1k , . . . , xmnk ]
is a convergent subsequence.
91
(B.2.28)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Before being able to prove the equivalence of compactness and sequential compactness in metric spaces several new concepts needs to be introduced.
First of these is the concept of countability. It deals with the cardinality of sets
and more specically it gives a way to categorize sets with innite cardinality
as countable or uncountable.
Denition B.2.3 (Countable).
A countable set X is a set with nite cardinality or a set for which there exists a bijection E.0.11 f : N → X between the
natural numbers and X , otherwise X is uncountable.
The following proposition gives a useful method to prove countability of a set
by a diagonalization process.
Proposition B.2.1. [Mor05]The set Q is countable.
Proof. The set Q is clearly not of nite cardinality so there has to be a bijection
between the natural numbers N and Q for Q to be countable. It is sucient
to show for the positive rational numbers that such a bijection exists because
otherwise just alternate between positive and negative number in this fashion
for a given list of positive rational numbers {ai }:
1 ↔ a0 , 2 ↔ −a0 , 3 ↔ a1 , 4 ↔ −a1 , . . . .
(B.2.29)
Arrange the positive rationals as in Figure 12.
Start the listing at the upper left corner with 1/1 and move through the diagonals as intended in the Figure 12 and skip new numbers in the list which has
already been repeated like, 1/1 = 2/2. This procedure will produce a bijection
of the form:
1 ↔ 1/1, 2 ↔ 2/1, 3 ↔ 1/2, 4 ↔ 3/1, 5 ↔ 1/3, 6 ↔ 4/1, 7 ↔ 3/2, 8 ↔ 2/3, 9 ↔ 1/4, . . . .
(B.2.30)
Remark B.2.4. The method used above in the proof of the countability of Q is
called the Cantor's diagonalization argument .
Example B.2.6 (Countable). The sets Z, 2Z, N, X = {0, 1, 2, 3, 4} are countable.
This proposition shows that R is not countable and hence the cardinality of Z,
Q is not equal to R.
Proposition B.2.2. [Mor05]The set R is uncountable
92
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Figure 12: A counting of the rationals
Remark B.2.5. Hence on a listing of a countable set will be a bijection as these
seen in the proof of Proposition B.2.1.
Proof. Assume R is countable. Then the positive reals need to be countable.
Take a listing of these. Construct an α such that it diers in the rst decimal
place from the rst decimal for the rst element in the list, such that it diers
in the second decimal place from the second decimal for the second element in
the list and so on. This α will not be listed and we have a contradiction. Then
the set R is uncountable.
The next needed concept is denseness. Subsets are dense if every point in the set
is an accumulation point. Below is a formalization of the concept of denseness.
Denition B.2.4 (Dense). A subset Y of a topological space X is
every x ∈ X , x belongs to Y or is an accumulation point of Y .
dense if for
Example B.2.7 (Dense). The subset Q of R is dense in R. To show that this
is the case, choose x ∈ R such that x ∈/ Q. But there is not an open set which
contains x but do not contain any points in Q. Therefore x is an accumulation
point of Q and Q is dense.
Example B.2.8 (Dense). The subset of irrational numbers, R \ Q is dense in
This shows that a set can have more then one dense subset and in the earlier
example the cardinality of Q is lower then of R but the cardinality of R\Q equals
that of R.
R.
93
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Example B.2.9 (Dense). The subset Q of R is not dense in the topological
space R with the discrete topology. This is easily realized when considering that
every subset of R is open. Choose x ∈ R such that x ∈/ Q. The open set x do
not contain any points in Q, so x is not an accumulation point and therefore Q
is not dense in R with the discrete topology.
The next concept is separability. A set is separable if it has a countable dense
subset.
Denition B.2.5 (Separable). A topological space X is separable if it contains
a countable dense subset.
Example B.2.10 (Separable). Given the space R. The subset Q is a countable
dense subset of R and hence R is separable.
The last topological property needed is the notion of a base for a topological
space. This concept is formalized below.
Denition B.2.6 (Base). A collection {Uα } of open sets in a topological space
X is a base for X if for every x ∈ X and every open set U ⊂ X with x ∈ U ,
there is an α such that:
(B.2.31)
x ∈ Uα ⊂ U.
Example B.2.11 (Base). Given the topological space R with the standard topology. The collection of open sets formed by the topology of R is a base for R.
Example B.2.12 (Base). Given the topological space R with the discrete topology. The collection of open sets formed by the topology of R is not a base for R.
To see this choose x ∈ R and the open set x. Now there is clearly not an open
set U such that x ∈ U ⊂ x.
The proof of the equivalence between sequential compactness and compactness
B.2.3 in metric spaces is divided into the four following lemmas and then the
main theorem.
The rst Lemma B.2.4 gives that every separable metric space has a countable
base.
Lemma B.2.4. [Rud76] Given a separable metric space
topology. Then there exists a countable base for X
X
and the standard
Proof. Using the denition of a separable space there is a countable dense subset:
C = {c1 , c2 , . . .}
94
(B.2.32)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Figure 13: An element of a base
of X . Consider the countable collection of open sets:
B = {Bd (ci , r) : r ∈ Q, i = 1, 2, . . .} .
(B.2.33)
What remains to show is that B is a base for X .
Choose x ∈ X and an open set U such that x ∈ U . Then from the denition
of open set in the standard topology B.1.6 there is an open ball Bd (x, r) such
that:
Bd (x, r) ⊂ U,
(B.2.34)
without loss of generality assume that r ∈ Q. From the denition of dense, x is
an accumulation point of C and hence the ball:
Bd (x, r/2)
(B.2.35)
contains at least one element of C . This says that there is an i such that
d(x, ci ) < r/2.
(B.2.36)
Bd (ci , r/2) ∈ B
(B.2.37)
This gives that the ball:
95
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
satises:
(B.2.38)
x ∈ Bd (ci , r/2) ⊂ U
and hence C is a base.
The next Lemma B.2.5 says that a sequential compact space is separable.
Lemma B.2.5. [Rud76] Given a sequentially compact space
separable.
X.
Then X is
Proof. Fix δ > 0 and x1 ∈ X . Now choose x2 ∈ X such that:
d(x1 , x2 ) ≥ δ.
(B.2.39)
Continue choosing x1 , x2 , . . . , xj+1 such that:
d(xi , xj+1 ) ≥ δ for i = 1, 2, . . . , j.
(B.2.40)
This gives a sequence with a distance of at least δ between each element. This
can not be done innitely because otherwise there would not exists a converging
subsequence of {xj } and this would give a contradiction because X is sequentially compact. Now choose j such that for every point in X , the distance to
some xi is less than δ . This is possible by choosing xj+1 ∈ X , which gives that:
d(xj+1 , xi ) < δ
(B.2.41)
for some i = 1, 2, . . . , j . Continue choosing xj+2 , xj+3 , . . . , xj+h until:
d(xk , xl ) < δ
(B.2.42)
for every pair xk , xl . This can be done by choosing xj+i in the following way: Let
r = max {min {d(xi , xj ) : for j = 1, 2, . . . , j + 1 and j 6= i}} .
i
(B.2.43)
Then take the xi with the minimum index such that:
min {d(xi , xj ) : for j = 1, 2, . . . , j + 1 and j 6= i} = r
(B.2.44)
and denote as xk . Finally choose xj+i ∈ Bd (xk , r). Now the nite set {xi }i=1,2,...,j+h
satises that for every point in X , the distance to some xi is less than δ . This
gives that:
[
Bd (xj , δ) = X.
(B.2.45)
j=1,2...,j+h
96
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
1
Let
δ
=
for
n
=
1,
2,
.
.
.
.
For
n
=
1
this
yields
the
sequence
x1j and n = 2
n
n
o
x 12 . The set:
j
o
n
B = x n1 : n = 1, 2, . . . , i = 1, 2, . . . , j
i
(B.2.46)
is countable. Choosing x ∈ X and a ball Bd (x, r). This ball contains a y ∈ X
such that y 6= x because for n1 < r there exists a x n1 such that:
i
d(y, x n1 ) < r,
i
(B.2.47)
hence B is dense in X and thus X is separable.
Remark B.2.6. The two Lemmas B.2.4, B.2.5 gives together that every sequentially compact space X has a countable base.
The next Lemma B.2.6 is of a technical nature and states that every open covering of a space with countable base has a countable subcovering.
Lemma B.2.6. [Rud76] Given a space X with countable base. Then every
open covering of X has a countable subcovering.
Remark B.2.7. The type of compactness in Lemma B.2.6 that every open covering of a space X has a countable subcovering is often called Lindelöf compactness .
Proof. Let {Vj } be a base for X and choose an open covering {Uα } of X . Now
let J be the set of all j such that there exists an α for which Vj ⊂ Uα .
The set {Vj }j∈J is an open covering of X which is seen by: choose x ∈ X , then
there exists an α such that x ∈ Uα . From the denition of a base there exists a
Vi such that:
x ∈ Vi ⊂ Uα .
(B.2.48)
Then i ∈ J and
x∈
[
Vj .
(B.2.49)
j∈J
Furthermore for each j ∈ J, choose αj such that Vj ⊂ Uαj . This gives that:
[
[
Vj ⊆
Uαj = X
(B.2.50)
j∈J
j∈J
and hence Uαj is a countable subcovering.
The last Lemma B.2.7 gives that the intersection of a sequence of sets is nonempty if every set is a subset of the previous sets in the sequence.
97
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Lemma B.2.7. [Rud76] Given a sequence of non-empty closed subsets {Xn }
of a sequentially compact set X such that X1 ⊃ X2 ⊃ . . .. Then:
∞
\
Xi 6= ∅.
(B.2.51)
i=1
Proof. Choose xn ∈ Xn for each n and consider the set:
A = {xn : n = 1, 2, . . .} .
(B.2.52)
If A is nite then there exists a xk which belongs to innitely many Xn . Because
X1 ⊃ X2 ⊃ . . . this gives that xk belongs to the intersection and hence is not
empty.
Thus assume that A is innite. Because X is sequentially compact, the sequence
A has a converging subsequence with a limit a ∈ X and hence is an accumulation point. Now for a x value of n, every neighborhood U around a contains
innitely many points of A and among these there is such that xi ∈ U for i ≥ n.
This gives that:
xi ∈ Xi ⊂ Xn .
(B.2.53)
Since every neighborhood of a contains a point of Xn , a is an accumulation
point of Xn and Xn is closed gives that a ∈ Xn . This reasoning applies for
every n and thus
∞
\
a∈
Xi .
(B.2.54)
i=1
Now nally Theorem B.2.3 is ready to be proven. This theorem is repeatedly
used in Chapter 2.
Theorem B.2.3. [Rud76] Given a metric space X . Then the following statements is fullled:
X
is a compact space ⇐⇒ X is a sequentially compact space
Proof. =⇒: Assume X is a compact space.
Assume that X is not sequentially compact. Now it is possible to choose a sequence S = {xk } ∈ X without a subsequence converging in X . Then S cannot
have an accumulation in X . This gives that there is an open covering {Ux }∀x∈X
of S such that each Ux has at most one point in common with S . But there
does not exists a nite subcovering of {Ux }∀x∈X which covers S , is not compact
and therefore X cannot be compact which gives a contradiction. Thus X is
sequentially compact.
⇐=: Assume X is a sequentially compact space.
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B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
From Lemmas B.2.4, B.2.5 gives that X has a countable base. Choose an open
covering {Uα } and by Lemma B.2.6, X is Lindelöf compact, i.e. every open
covering has a countable subcovering. Let {Ui } be such a covering. If {Ui } is
nite the proof is complete, hence assume {Ui } is innite and thus
U1 ∪ U2 ∪ . . . ∪ Un + X
(B.2.55)
for every n.
Now let
Xn = {x ∈ X : x ∈
/ U1 ∪ U2 ∪ . . . ∪ Un } = X ∩ U1c ∩ U2c ∩ . . . Unc .
Because
U1 ∪ U2 ∪ . . . ∪ Un
(B.2.56)
(B.2.57)
is open, the set Xn is closed due to Lemma B.2.1 and furthermore X1 ⊃ X2 ⊃ . . .
so using Lemma B.2.7 gives that:
∞
\
Xi 6= ∅
(B.2.58)
i=1
and thus there exists a y such that:
∞
[
y∈
/
Ui
(B.2.59)
i=1
which is a contradiction because {Ui } is a covering of X . This gives that {Ui }
is a nite subcovering a X is compact.
The next Theorem B.2.9 is needed in the Lebesgue´s lemma but before proving
this two lemmas will be stated and proved.
The rst lemma gives that the image of a continuous function on a compact set
is compact.
Lemma B.2.8. [Mor05] Given compact set X and a continuous function f .
Then the image of f is a compact set.
Proof. Choose an open covering {Uα } of f [X]. Because f is continuous, f −1 [Uα ]
is an open set. Considering the union:
[ [
−1
−1
f [Uα ] = f
Uα = f −1 [f [X]] = X
α
(B.2.60)
α
and thus
f −1 [Uα ]
99
(B.2.61)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
is an open covering of X . Because X is compact there exists a nite subcovering
n
of (B.2.61), {U }i=1 . Rewriting f [X] gives:
f [X] = f [
n
[
[
n
n
[
f −1 [Ui ]] = f f −1 [ Ui ] ⊆
Ui
i=1
i=1
(B.2.62)
i=1
n
which gives that {U }i=1 is a nite subcovering of f [X] and thus f [X] is compact.
This Lemma B.2.9 states that a real function on a compact set has a maximum
and a minimum.
Lemma B.2.9. [Mor05] Given a compact set X and a continuous function
Then f has a maximum and a minimum.
f : X → R.
Proof. From Lemma B.2.8 it follows that f [X] is compact and using Heine-Borel
Theorem B.2.1 gives that f [X] is closed and bounded. Let a = sup f [X] and
b = inf f [X]. Because f [X] is closed, a and b belongs to f [X] and are thus
maximum and minimum of f [X] respectively.
Remark B.2.8. A continuous function f : X → R on a compact set X has a
maximum if there exists an element f (x0 ) such that:
f (xo ) ≥ f (x) for every x ∈ X
and a
(B.2.63)
minimum if there exists an element f (x0 ) such that:
f (xo ) ≤ f (x) for every x ∈ X.
(B.2.64)
The Lebesgue´s Lemma B.2.4 gives that there is always possible to nd a ball
with positive radius which is contained in an open set belonging to an open
covering. This theorem is of a technical nature and is used in Chapter 2.
Theorem B.2.4 (Lebesgue´s Lemma). [BBI01] Given a compact metric space
X . Let {Uα } be an open covering of X . Then there exists a ρ > 0 such that
any ball of radius ρ is contained in some of the sets Uα .
Remark B.2.9. The number ρ in Theorem B.2.4 is often called Lebesgue
of the covering.
number
Proof. The metric of X is nite because otherwise it is not possible to create an
open covering. Further assume that none of the Uα covers the whole X because
otherwise the proof is completed. Then create the function f : X → R, where
f (x) = sup {r ∈ R : Bd (x, r) is contained in some Uα } .
100
(B.2.65)
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
This function is well-dened because {Uα } is an open covering and f is positive
for every x ∈ X and it is continuous because f is Lipschitz continuous, which
is seen by noting that
(B.2.66)
d(f (x1 ), f (x2 )) ≤ d(x1 , x2 )
and thus continuous by Theorem A.0.9. Then using Lemma B.2.9 gives that f
has a minimum a. Now let ρ = a/2.
Curves plays an important part of the theory in this thesis and of especial importance in Chapter 2. The reader should check the dierence between curves
and paths (see Denition 2.1.1).
Denition B.2.7
(Curve). Given a topological space X . A
tinuous map γ : I → X , where I is an interval.
curve γ is a con-
Example B.2.13 (Curve). Given the topological space
topology. The continuous map:
with the standard
(
x = t,
γ(t) =
y = t2 ,
R2
t ∈ I[0,3]
,
t ∈ I[0,3]
(B.2.67)
where I[0,3] is the closed interval between 0 and 3, and thus γ(t) is a curve in R2 .
Example B.2.14 (Curve). Given the topological space R with the discrete topology. The map:
(
γ(t) =
t∈Q
t∈R\Q
1,
0,
(B.2.68)
is continuous so γ is a curve.
In Chapter 2 shortest paths and geodesics will play a central part but to be able
to dene these there is needed some kind of method to measure the length of a
curve and this is done in the following way.
Denition B.2.8 (Length of curve).
Given a topological space X with metric
d. The length of a curve γ : I[ a, b] → X with a partition:
a = t1 < t2 . . . < tn = b
(B.2.69)
is given by:
l(γ) =
sup
n−1
X
d(γ(ti ), γ(ti+1 ))
(B.2.70)
a=t1 <t2 ...<tn =b i=1
Here the supremum is taken over all possible partitions of I[ a, b] and n is unbounded. The curve length l is called rectiable if l is nite.
101
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Remark B.2.10. Two curves γ1 and γ2 with equal images γ1 (I[ a, b]) and γ2 (I[ c, d])
have the same length.
Almost all spaces in this thesis will be what is called Hausdor spaces. A Hausdor space in a metric space is essentially a space in which there is a positive
distance between every distinct points in the space and this is generalized in the
following matter.
Denition B.2.9
(Hausdor space). A topological space is a Hausdor space
X if every two distinct points are separable by neighborhoods, i.e. for given
points x and y there exist neighborhoods U ,V to x, y respectively such that:
(B.2.71)
U ∩ V = ∅.
Example B.2.15 (Hausdor space). Given the topological space (Z, σ) where
σ is the discrete topology. Then any members a, b ∈ Z is separable by the neighborhoods U = {a} and V = {b} respectively.
Example B.2.16 (Hausdor space). Given the topological space (R, σ) where
σ is the discrete topology. Then any members a, b ∈ R is separable by the neighborhoods U = {a} and V = {b} respectively.
Example B.2.17 (Hausdor space). Given the topological space (R, σ) where
σ is the trivial topology σ = {∅, R}. This space is not a Hausdor space which
can been seen by none of the members are separable by neighborhoods of the open
sets {∅, R}. For example choose x, y ∈ R. Neither x nor y belongs to ∅, so x, y
most be separated by neighborhoods U, V , where R ⊆ U and R ⊆ V . But this is
not true due to U ∩ V 6= ∅.
Denition B.2.10 (Domain). Given a nite
domain Ω is any connected open subset of X .
dimensional vector space X . A
Example B.2.18 (Domain). Given the vector space R. The set
{x ∈ R : 0 < x < 1}
(B.2.72)
is an open set in R and connected, so this set is a domain in R.
Example B.2.19 (Domain). Given the vector space R. The set
{x ∈ R : 0 < x < 1} ∩ {x ∈ R : 2 < x < 3}
(B.2.73)
is an open set in R but it is not connected, so this set is not a domain in R.
102
B.2 Compactness
B TOPOLOGICAL SPACES AND TOPOLOGY
Example B.2.20 (Domain). Given the vector space R2 . The set:
X = Bd (0, r) \ R+
(B.2.74)
is an open set in R2 .
A convex set in Rn is set which satises that every points can be joined by a
straight line segment.
Denition B.2.11 (Convex set). Given the space Rn . A subset X ⊂ Rn is
a convex set if every pair of points (x, y) ∈ X can be joined by a straight line
segment γ such that γ ∈ X .
Figure 14: A non convex set
Example B.2.21 (Convex set). Every ball in the Euclidean space E n is a convex space but no member of the set of sets constructed by taking the set of balls
in E n and removing the center point are a convex space. This can be seen by
the example of the unit ball B centered at [0, 0] in E 2 with [0, 0] removed and
choosing the points x = [0.5, 0], y = [−0.5, 0]. Then the line segment between x
and y is not in B because the line segment contains [0, 0].
103
C ALGEBRA
C
Algebra
The eld of algebra deals with operations and relations between sets. The rst
example of such operations are addition and multiplication of numbers which
are taught in elementary school. Generalizing this concepts gives rise to the eld
of abstract algebra where the objects groups, rings and elds are the concepts
of main focus. In this thesis the theory surrounding elds will be used covertly
in that sense that a lot of maps will be mapped into elds and thus a basic
understanding of elds is useful for understanding the material in mainly Chapter 3 and 4. For those readers wanting a more thorough examination of this
eld a basic textbook in abstract algebra as for example [Fra67] is recommended.
An equivalence relation is a relation on a set which relates if two elements in
the set are "equivalent" or not. In order to dene this strictly, the concepts of
Cartesian products and binary relations are needed.
Denition C.0.12
(Cartesian product). The
sets X and Y is dened by:
Cartesian product between two
X × Y = {(x, y)|x ∈ X, y ∈ Y } .
(C.0.75)
Example C.0.22 (Cartesian product). Given the sets X = {0, 1} and
Y = {1, 2}. The Cartesian product of X and Y is given by:
X × Y = {[0, 1], [0, 2], [1, 1], [1, 2]} .
(C.0.76)
Denition C.0.13 (Binary relation). A binary relation R is an ordered triplet
(X, Y, G), where X and Y are sets and G is a subset of the Cartesian product
X × Y . Here X is the relations domain, Y the codomain and G the graph.
Remark C.0.11. A binary relation (X, X, G) is a binary operator if G satises:
G : X × X → X.
(C.0.77)
This property makes a binary operator closed.
Example C.0.23 (Binary relation). One binary relation R = (X, Y, G) is given
by:
R = (X = P, Y = N, G = {[2, 0], [2, 2], [2, 4], ...[3, 0], [3, 3], [3, 6], ...}), (C.0.78)
where P is the set of primes, N is the set of natural numbers and G is constructed
by a member p ∈ P is associated with every member of N which is a multiple of p.
105
C ALGEBRA
Denition C.0.14 (Equivalence relation). Given graph ∼ on a set X (in Definition C.0.13 we have the following triplet (X, X, ∼)). ∼ is an equivalence
relation if the following is satised for all x, y, z ∈ X :
a) x ∼ x (Reexive)
b) x ∼ y =⇒ y ∼ x (Symmetric)
c) x ∼ y and y ∼ z =⇒ x ∼ z (Transitive)
Remark C.0.12. Two elements a and b satises a ∼ b if the element [a, b] of the
Cartesian product belongs to the graph ∼.
Example C.0.24 (Equivalence relation). One example of an equivalence relation on Z is equal to (a = b). Using the terminology in Denition C.0.14 we
have the triplet (Z, Z, G), where
G = {[0, 0], [1, 1], [−1, −1], [2, 2], [−2, −2], ...} .
(C.0.79)
Example C.0.25 (Equivalence relation). Another example of an equivalence
relation on Z is modulus (a = b mod(n)). The triplet is now (Z, Z, G) for n = 3,
where
G = {[0, 0], [0, 3], [0, −3], [0, 6], [0, −6], ..., [1, 1], [1, 4], [1, −2], [1, 7], [1, −5], ...} .
(C.0.80)
Example C.0.26 (Equivalence relation). One example of a relation which is
not an equivalence relation on Z is greater than (≤). It satises the reexivity
and the transitivity but it is not symmetric. For example 0 ≤ 1 but this does not
imply 1 ≤ 0.
An equivalence class to a specic element is then the natural extension of letting
every equivalent element to the specic element belong to the same equivalence
class.
Denition C.0.15 (Equivalence class). Given a set
relation ∼. The equivalence class of x0 ∈ X is the set:
[x0 ]X = {x ∈ X : x0 ∼ x} .
Example C.0.27
3 for the set N is:
(Equivalence class).
X and an equivalence
(C.0.81)
The equivalence class of 1 for modulus
[1]N = {1, 4, 7, 10, ...} .
106
(C.0.82)
C ALGEBRA
In abstract algebra, the rst object often considered is a group. A group is a
set together with an operation dened on this set. This operation shall satisfy
the conditions of closure, associativity, existence of identity and inverse element.
The meaning of these conditions will be claried in the following denition but
one reason for this abstract denition is to get a generalization of the properties
of for example addition on the set of integers.
Denition C.0.16
(Group). A
satises for all a, b, c ∈ G:
group is a set G and an operation (•) that
a) a • b ∈ G (Closed)
b) (a • b) • c = a • (b • c) (Associative)
c) There exists an element e ∈ G such that:
e • a = a • e = a (Identity element)
(C.0.83)
d) To each a there exists an element a−1 ∈ G such that:
a • a−1 = a−1 • a = e (Inverse element)
(C.0.84)
Example C.0.28 (Group). The set Z under addition (+) is a group. Here the
identity element is given by 0 and for example the inverse element of an element
a ∈ Z is given by −a.
Example C.0.29 (Group). The set Z under multiplication (·) is not a group.
The identity element exists and is given by 1 but the inverse element only exists
for the element 1.
Example C.0.30 (Group). The set X = {0, 1} under addition and modulo 2
is a group. Here the identity element is 0 and the inverse element of 1 is 1 and
0 is 0.
When a group is dened, an abelian group is simple a group which is commutative.
Denition C.0.17
if for all a, b ∈ G:
(Abelian group). A group (G, •) is
a • b = b • a.
abelian (commutative)
(C.0.85)
Example C.0.31 (Abelian group). The group in Example C.0.28 is an Abelian
group due to the addition operator being abelian, i.e. a + b = b + a.
107
C ALGEBRA
Example C.0.32 (Abelian group). The set Z under the operation subtraction
(−) is not an abelian group. This is due the operator not being abelian, for
example 0 − 1 6= 1 − 0 and not associative, (1 − 2) − 3 = −4 and 1 − (2 − 3) = 2.
A eld is an extension of a group by which instead of only dening one operation
on the set, two operations are dened and not only shall the set together with
on of the operations be a group (not strictly true, see Denition C.0.18) but
also the two operations need to satisfy the distributive law. It is strictly dened
in the following way.
Denition C.0.18 (Field). A eld is a set F together with two operations
(+), (•) (denote the operations addition and multiplication) that satises for all
a, b, c ∈ F :
a) (F, +) is an abelian group
b) (F \ {0} , •), is an abelian group where 0 is the additive identity element
c) a • (b + c) = (a • b) + (a • c) (Distributive)
Example C.0.33 (Field). The set R with the two operations additive addition
and multiplication (·) is a eld. (R, +) is an abelian group with identity
element 0 and the inverse element of a ∈ R is −a. The set (R \ {0} , ·) is also
an abelian group with identity element 1 and inverse element to a is a1 . The
distributivity for + and · is also fullled because:
(+)
a · (b + c) = (a · b) + (a · c).
(C.0.86)
Example C.0.34 (Field). The set of rational numbers Q with the two operations addition (+) and multiplication (·) is a eld. (Q, +) is an abelian group
with identity element 0 and the inverse element of ab ∈ Q is − ab . The set
(Q \ {0} , ·) is also an abelian group with identity element 1 and inverse element
to ab is ab .The distributivity for + and · is also fullled because:
a c
e
a c
a e
· ( + ) = ( · ) + ( · ).
b d f
b d
b f
(C.0.87)
Example C.0.35 (Field). The set Z with the two operations addition (+) and
multiplication (·) is not a eld. (Z, +) is an abelian group with identity element
0 and the inverse element of a ∈ Q is −a. But neither (Z, ·) nor (Z \ {0} , ·) is
an abelian group due to the inverse element does not exists except for 1 ∈ Z.
When a eld is dened, a subeld is a subset which contains the two identity
elements and also is a eld.
Denition C.0.19 (Subeld). A subeld F1 of a eld F is a subset that contains the additive identity element 0 and the multiplicative identity element
denoted as 1, and such that F1 is also a eld.
108
C.1 Linear algebra
C ALGEBRA
Example C.0.36 (Subeld). The eld (Q, +, ·) is a subeld of the eld (R, +, ·).
Remark C.0.13. Observe that Z is not a subeld of R or Q.
Denition C.0.20 (Permutation).
Given a set X . A
tion (see Denition E.0.11) σ : X → X .
Example C.0.37 (Permutation). Given the set
tion σ of X is given by:
σ=
1
2
2
1
3
4
4
3
permutation σ is a bijec-
X = {1, 2, 3, 4}.
A permuta(C.0.88)
where the above notation means that 1 is mapped on 2, 2 is mapped on 1, 3 is
mapped on 4 and 4 is mapped on 3.
C.1 Linear algebra
The goal of this section is to give the theory necessary for Sylvester's criterion
which is a condition when a matrix is positive-denite. This will be needed in
Chapter 4. For a more complete exposition of this subject see [HJ85].
The determinant is a value associated with an n × n-matrix. Several dierent
(and equivalent) denitions exist of determinants and one of these is Denition
C.1.1 stated below. Note that the matrix specied in the denition is not necessarily real-valued nor complex-valued but instead only need to belong to a elds
such that the operations + and · are specied.
Denition C.1.1 (Determinant). Given an n × n matrix A. Let Σ be the set
of all permutations σ of {1, 2, . . . , n}. Then the determinant of A, det(A) is
given by:
det(A) =
X
sgn(σ)
σ∈Σ
n
Y
ai,σ(i)
(C.1.1)
i=1
where sgn(σ) is dened as 1 if σ does an odd number of reorderings of {1, 2, . . . , n}
and −1 if σ does an even number of reorderings, for example
1 2 3 4
σ=
,
(C.1.2)
2 1 4 3
σ does an even number (2) of reorderings so sgn(σ) = −1. The entity ai,σ(i)
is the element of the matrix A which corresponds to the i:th row and σ(i):th
column.
109
C.1 Linear algebra
C ALGEBRA
Example C.1.1 (Determinant). Given a
terminant of A is then given by:
P
σ∈Σ
sgn(σ)
2
Q
ai,σ(i) = sgn([1, 2])
i=1
= a1,1 a2,2 − a1,2 a2,1 = ad − bc.
2
Q
2×2
matrix
ai,σ(i) + sgn([2, 1])
i=1
a
A=
c
2
Q
b
.
d
The de-
ai,σ(i) =
i=1
A complex matrix is dened as positive-denite if the following is satised:
Denition C.1.2
(Positive-denite). Given a matrix A where each element
aij ∈ C. The matrix A is positive-denite if
(C.1.3)
z T Az > 0
for every non-zero vector z ∈ C.
Example C.1.2 (Positive-denite). Given the matrix A =
a, b ∈ R+ . Then A is positive-denite matrix because choosing
gives:
0 a
, where
b 0
z = [z1 , z2 ] 6= 0
(C.1.4)
z T Az = 0 + z 2 z2 b + z 1 z1 a + 0 = Re(z22 )b + Re(z12 )a > 0.
Example C.1.3 (Positive-denite). Given the matrix

2
A = −1
0
−1
2
−1

0
−1
2
(C.1.5)
and the vector z = [z1 , z2 .z3 ]. This gives for zj = xj + iyj :
 
z1
(−z 1 + 2z 2 − z 3 ) (−z 2 + 2z 3 ) z2  =
z3
= 2z 1 z1 − z 2 z1 − z 1 z2 + 2z 2 z2 − z 3 z2 − z 2 z3 + 2z 3 z3 =
z T Az = (2z 1 − z 2 )
= 2Re(z12 ) − 2(x1 x2 + y1 y2 ) + 2Re(z22 ) − 2(x2 x3 + y2 y3 ) + 2Re(z32 ) =
= Re(z12 ) + Re(z32 ) + (x1 − x2 )2 + (y1 − y2 )2 + (x2 − x3 )2 + (y2 − y3 )2
and thus z T Az > 0 if z 6= 0. This example shows that matrices with negative
valued elements can be positive-denite.
Example C.1.4 (Positive-denite). Given the matrix
vector z = [1, −1], gives that:
z T Az = −3
110
A =
1
2
3
1
and the
(C.1.6)
C.1 Linear algebra
C ALGEBRA
This shows that it is not a sucient condition for a matrix to have positive
elements for it to be positive-denite.
A complex matrix is an Hermitian matrix if it equal to its conjugate transpose.
Denition C.1.3 (Hermitian matrix). A matrix A is an
for each element aij of A:
aij = aji .
Hermitian matrix if
(C.1.7)
Example C.1.5 (Hermitian matrix). The matrix:
π
3 − 2i
3 + 2i
17
21
(C.1.8)
is an Hermitian matrix.
The Sylvester`s criterion gives the equivalence of positive deniteness and that
every upper left corner submatrix has a positive determinant. This gives in
some cases an easy method of checking whether a matrix is positive-denite or
not.
Theorem C.1.1 (Sylvesters criterion). Given an Hermitian n × n matrix A.
Then the following conditions are equivalent:
a) A is positive-denite
b) The following matrices has positive determinants:
The upper left 1-by-1 corner of A
The upper left 2-by-2 corner of A
..
.
The upper left n-by-n corner of A
Proof. The proof of this theorem is omitted here but can be found in [Gil91]
111
D VECTOR SPACES
D
Vector spaces
In this Appendix, vector spaces together with some concepts often associated
with vector spaces such as inner product, norms, basis, dual spaces, tensors and
dierential forms will be dened and be given examples of. Most of the results
and concepts in this appendix will be used in Chapter 3 and Chapter 4. To the
reader who wish to study these subjects closer the books [AR10] and [Car06]
are recommended.
A vector space is a eld whose elements are called vectors and satisfying the eight
conditions in Denition D.0.4. One important vector space which is studied in
this thesis is the tangent space Denition 3.1.14. The additive and multiplicative properties among others makes vector spaces easy to work with.
Denition D.0.4
(Vector space). A vector space (linear space) over a eld F
is a set X with two binary operators +, · where (+) operates between elements
of X and (·) between an element of F and X . For X to be a vector space the
following is satised for every u, v, w ∈ X and a, b ∈ F
a) u + (v + w) = (u + v) + w (Associative)
b) u + v = v + u (Commutative)
c) There exists an element 0 ∈ X called the zero vector such that:
v + 0 = 0 + v = v (Additive identity element)
(D.0.9)
d) There exists an additive inverse −v ∈ X such that:
v + (−v) = −v + v = 0 (Inverse element)
(D.0.10)
e) a · (u + v) = (a · u) + (a · v) (Distributive)
f) (a + b) · v = (a · v) + (b · v) (Distributive)
g) a · (b · v) = (a · b) · v (Associative)
h) There exists an element 1 ∈ F such that 1 · v = v (Multiplicative identity)
Remark D.0.1. The condition that +, · are binary operators insures that:
u + v ∈ X and a · u ∈ X.
(D.0.11)
Example D.0.6 (Vector space). The set R with the operations addition (+)
and multiplication (·) over the eld R is a vector space. For example the zero
vector is 0, the multiplicative identity element is 1 and the additive identity element is 0.
113
D VECTOR SPACES
A linear map is a function, mapping elements of a vector space to another vector
space such that the function is both additive and homogeneous.
Denition D.0.5 (Linear map). Given two vector spaces X1 and X2 over a
eld F . A linear map L is a function L : X1 → X2 that satises for all x, y ∈ X1
and c ∈ F :
a) L(x + y) = L(x) + L(y) (Additive)
b) L(cx) = c · L(x) (Homogeneity)
Example D.0.7 (Linear map). Given the vector space R with the operations
addition (+) and multiplication (·) over the eld R. Then the map L : R → R,
L(x) = cx is a linear map. This can be seen for all x, y, c, d ∈ R by:
a) L(x + y) = c(x + y) = cx + cy = L(x) + f (y)
b) L(dx) = dcx = d · L(x)
A basis for a vector space is a subset of the vector space which is linearly independent and spans the whole vector space and by which means:
Denition D.0.6 (Basis).
A basis E = {v1 , ..., vn } of a vector space X over a
eld F is a nite subset of X which satises for all
a1 , ..., an ∈ K and v1 , ..., vn ∈ V :
(D.0.12)
a) If a1 v1 + ... + an vn = 0 =⇒ a1 , ..., an = 0 (Linearly independent)
b) For every x ∈ X it is possible to choose a1 , ..., an ∈ K such that:
x = a1 v1 + ... + an vn . (Spanning)
(D.0.13)
Example D.0.8 (Basis). Consider the vector space R2 (see Denition A.0.8)
with the operations addition (+) and multiplication (·) over the eld R2 . The
vectors e1 = [1, 0] and e2 = [0, 1] form a basis for R2 . This basis is called the
standard basis. Other vectors x, y satisfying being basis for R2 are for example
x = [1, 2] and y = [2, 1].
Example D.0.9 (Basis). Given the vector space in Example D.0.8. The vectors x = [1, 2], y = [2, 4] are not a basis for R2 . It is easily visible that x, y are
linearly dependent by: 2x − y = 0.
For every given vector space a corresponding dual space is dened as the set of
every linear map from the vector space to the corresponding eld. In this thesis
many objects in Chapter 3 and Chapter 4 will belong to dual spaces so dual
spaces will be an important concept.
114
D VECTOR SPACES
Denition D.0.7 (Dual space). Given a vector space X over a eld F . The
dual space X ∗ to X is given by the set of all linear maps (linear functionals)
f : X → F.
Example D.0.10 (Dual space). The dual space of the vector space (R, +, ·)
over the eld R is the maps:
{f : f (x) = cx, c, x ∈ R} .
(D.0.14)
That in fact all linear maps in Example D.0.10 are of the form f (x) = cx can
be seen by the following reasoning:
Let g be a linear map g : R → R of the vector space (R, +, ·) over the eld R.
Then:
g(x) = g(x · 1) = x · g(1)
(D.0.15)
is of the form:
f (x) = cx
(D.0.16)
due to the fact that g(1) ∈ R.
Then the dual basis is simply dened as:
Denition D.0.8
basis for X ∗ .
(Dual basis). Given a vector space W . A
dual basis is a
Example D.0.11 (Dual basis). The basis for the dual space in Example D.0.10
is any member of R\ {0}.
Many geometrical objects are tensors. For example scalars, vectors and linear maps are examples of dierent tensors. Two of the rst to study what
is today known as tensors where Ricci and Levi-Civita during the late 19:th
century when developing the theory surrounding the curvature of Riemannian
manifolds. There are several dierent equivalent denitions of tensors used in
dierent books and the one used in this thesis denes a tensor as a multilinear
mapping satisfying the following condition:
Denition D.0.9 (Tensor).
tensor is a map:
Given a vector space X over a eld F . An (n × m)
T : X∗ × X∗ × . . . × X∗ × X × X × . . . × X → F
|
{z
} |
{z
}
n times
m times
which is linear in every argument (multi linear).
115
(D.0.17)
D VECTOR SPACES
Example D.0.12 (Tensor). Given the vector space Rn and
(D.0.18)
x = [x1 , . . . , xn ] ∈ Rn .
Then the mean function:
E : Rn → R, E(x) =
n
X
xi
i=1
n
(D.0.19)
is a tensor.
To be able to talk about length of vectors and orthogonality in vector spaces an
inner product needs to be dened. The inner product is a mapping which takes
a pair of vectors and maps them on the scalars.
Denition D.0.10 (Inner product). Given a subeld F of the complex numbers
C and a vector space X . An inner product h·, ·i is a map:
h·, ·i : X × X → F
(D.0.20)
which satises for all x, y, z ∈ X and a ∈ F :
a) hx, yi = hy, xi (Conjugate symmetric)
b) hax, yi = ahx, yi and hx + y, zi = hx, zi + hy, zi (Linear)
c) hx, xi ≥ 0 and if hx, xi = 0 =⇒ x = 0 (Positive-denite)
Example D.0.13 (Inner product). Consider the vector space R over the eld
R with the operations addition (+) and multiplication (·). The map:
(D.0.21)
hx, yi = xy
is an inner product which is easily visible by:
a) hx, yi = xy = yx = hy, xi = hy, xi.
b) hax, yi = axy = ahx, yi and
hx + y, zi = (x + y)z = xz + yz = hx, zi + hy, zi
c) hx, xi = x2 ≥ 0 and if hx, xi = x2 = 0 =⇒ x = 0
Example D.0.14 (Inner product). The inner product in Denition F.0.12 for
where x = [x1 , ..., xn ] and y = [y1 , ..., yn ] are given by:
x, y ∈ X ,
hx, yi = x · y =
n
X
xi yi
(D.0.22)
i=1
This is a generalization of the inner product in Example D.0.13 for n-dimensions:
a) hx, yi = x · y = y · x = hy, xi = hy, xi.
116
D VECTOR SPACES
b) hax, yi =
n
P
axi yi = a
i=1
hx + y, zi =
=
n
P
n
P
hx, xi =
and
(xi + yi )zi =
xi zi +
c) hx, xi =
xi yi = ahx, yi
i=1
i=1
n
P
i=1
n
P
yi zi = hx, zi + hy, zi
i=1
n
P
i=1
n
P
i=1
x2i ≥ 0
and if
x2i = 0 =⇒ x = 0
Example D.0.15 (Inner product). Consider the vector space R over the eld
R with the operations addition (+) and multiplication (·). The map:
< x, y >= x + y
(D.0.23)
is not an inner product because:
a0) hx, yi = x + y = y + x = hy, xi = hy, xi
b0) hax, yi = (ax) + y 6= ahx, yi and
hx + y, zi = (x + y) + z 6= hx, zi + hy, zi
c0) hx, xi = 2x 0
and if hx, xi = 2x = 0 =⇒ x = 0
So hx, yi = x + y satises a) but not b) nor c) in Denition D.0.10.
Naturally, an inner product space is then a vector space with a dened inner
product.
Denition D.0.11 (Inner product space).
An inner product space X is a vector space over the subeld F of C with an inner product h., .i.
Example D.0.16 (Inner product space). The vector space R over the eld
with the inner product for x, y ∈ R, hx, yi = xy is an inner product space.
R
The following theorem proves a useful inequality named the Cauchy-Schwarz
inequality. It is useful in many branches in mathematics and probability theory
and will be used a couple of times in proofs and examples in this thesis.
Theorem D.0.2 (Cauchy-Schwarz inequality). Given the inner product space
X and elements x, y ∈ X .
The following is satised: |hx, yi|2 ≤ hx, xi · hy, yi and is called the CauchySchwarz inequality.
117
D VECTOR SPACES
Proof. Given the inner product space X , elements x, y ∈ X and t ∈ R. Assume
hx, yi ∈ R.
Consider: htx + y, tx + yi = htx, tx + yi + hy, tx + yi =
= htx + y, txi + htx + y, yi =
= t2 hx, xi + thx, yi + t · hx, yi + hy, yi (by using t ∈ R)
= t2 hx, xi + 2t · Re(hx, yi) + hy, yi (by using hx, yi ∈ R)
= t2 hx, xi + 2t · hx, yi + hy, yi
This is a quadratic equation in t of type
at2 + bt + c
(D.0.24)
a = hx, xi, b = 2 · hx, yi
(D.0.25)
where
and c = hy, yi. Furthermore from the denition of inner product D.0.10:
htx + y, tx + yi ≥ 0,
(D.0.26)
at2 + bt + c ≥ 0 and a, c ≥ 0, b ∈ R
(D.0.27)
so:
Assume a 6= 0 and considering (D.0.27) yields:
at2 + bt + c ≥ 0 ⇐⇒ t2 + ab t +
⇐⇒ (t +
b 2
2a )
≥ − ac +
b2
4a2
c
a
=
≥ 0 ⇐⇒
b2 −4ac
4a2
b
Setting t = − 2a
which minimize (t +
b 2
2a )
yields:
b2 − 4ac ≤ 0
(D.0.28)
So applying (D.0.28) yields:
4|hx, yi|2 − 4hx, xihy, yi ≤ 0 ⇐⇒ |hx, yi|2 ≤ hx, xi · hy, yi
(D.0.29)
If a = 0, the equation (D.0.27) is
bt + c ≥ 0
(D.0.30)
which implies, b = 0 is the only solution for all t due to c ≥ 0. This solution
also satises (D.0.28), so the solution of (D.0.27) is (D.0.28).
118
D VECTOR SPACES
Assumehx, yi ∈ C. Choose λ such that |λ| = 1 and
hx, λyi = |hx, yi|
(D.0.31)
Considering: |hx, yi|2 = hx, λyi2 = (Using hx, λyi ∈ R)
= |hx, λyi|2 ≤ (Using Cauchy-Schwarz inequality for real valued inner products)
≤ hx, xihλy, λyi = hx, xi · (λλ)hy, yi = hx, xihy, yi
In vector spaces, norms are functions which assign lengths to vectors, that is a
non-negative number for each vector and only zero for the zero vector. Further
conditions for a function to be a norm on a vector space is given below.
Denition D.0.12 (Norm). Given a vector space X over a subeld F of C. A
norm is a function k·k : V → F such that for all x, y ∈ V and a ∈ F :
a) kaxk = |a| · kxk (Positive homogeneous)
b) kx + yk ≤ kxk + kyk (Triangle inequality)
c) If kxk = 0 =⇒ x = 0 (Zero vector)
The following Theorem D.0.3 gives an easy method for creating a norm from an
inner product.
Theorem D.0.3. Given an inner product space X over the eld F with inner
product h·, ·i. Then the following is a norm for x, y ∈ X and a ∈ F :
kxk =
Proof. For kxk =
D.0.12.
a) kaxk =
p
p
p
hx, xi
(D.0.32)
hx, xi to be a norm it needs to satisfy a)-c) in Denition
hax, axi =
p
p
aahx, xi = |a| hx, xi
b) To show the triangle inequality consider:
kx + yk2 = hx + y, x + yi = hx, x + yi + hy, x + yi =
= kxk2 + hx, yi + hy, xi + kyk2 ≤
≤ kxk2 + 2|hx, yi| + kyk2 ≤ (using the Cauchy-Schwarz inequality)
≤ kxk2 + 2kxk · kyk + kyk2 = (kxk + kyk)2
Taking the square rot of this expression shows b).
119
D.1 Dierential forms
D VECTOR SPACES
c) Using the denition of inner product D.0.10 and square rot mapping 0 on 0
proves c).
Remark D.0.2. In an inner productpspace with inner product h·, ·i, the norm
which will always be used is: kxk =
hx, xi.
Example D.0.17 (Norm). Given the vector space Rn in Denition A.0.8
√ over
the eld R. The norm dened in Denition F.0.12 for x ∈ Rn , kxk = x · x is
a norm given by the inner product D.0.14 satisfying Theorem D.0.3.
Example D.0.18 (Norm). Given the vector space R over the eld R. The
function f (x) = x, x ∈ R is not a norm for x, y, a ∈ R.
a0) f (ax) = ax = a · f (x)
b0) f (x + y) = x + y = f (x) + f (y)
c0)
If f (x) = x = 0 =⇒ x = 0
So f satises b) and c) in Denition D.0.12 but not a).
D.1 Dierential forms
Dierential forms is a subject studied, among others, the elds dierential geometry and tensor analysis. Élie Cartan is usually counted as one of the pioneers
in this subject due him publishing the paper "Sur certaines expressions diérentielles et le problème de Pfa" [Car99] in 1899 giving a more formal treatment of
dierential forms. Perhaps the rst dierential forms one encounter and a good
example are expressions of the form f (x)dx where f (x) is an integrand. The
layout and material in this section of the appendix follow close that of [Car06]
so for those readers interested in a more in depth study of dierential forms this
book is a good start.
Before dening dierential forms a couple of others concepts needs to be dened.
After dening dierential forms, the operation exterior product will be dened
for dierential for together with some rules of calculation.
The rst needed concept is alternating forms. One possible denition of alternating forms is given below:
Denition D.1.1 (Alternating multilinear form).
multilinear form is a map:
over a eld F . A
Given vector spaces X1 , . . . , Xn
f : X1 × . . . × Xn → F
(D.1.1)
which satises, for each i, the map:
f (x1 , . . . , xi , . . . , xn )
120
(D.1.2)
D.1 Dierential forms
D VECTOR SPACES
is a linear map of xi if all but the variable xi is constant and a map with n
variables which is a multilinear form is an n-linear form.
If the spaces X1 , . . . , Xn are equal, the map f is an
alternating map if
(D.1.3)
f (x1 , . . . , xn ) = 0
whenever xi = xi+1 for some 1 ≤ i < n.
Example D.1.1 (Alternating multilinear form). Given an n × n matrix A.
The determinant det(A) is an alternating multilinear form. This can be seen
by letting every column ci of A correspond to the vector space X over a eld F .
Then the map:
(D.1.4)
X × ... × X → F
given by the determinant is a multilinear map and an alternating map which
can be seen from the dening equation of a determinant (see Denition C.1.1).
det(A) =
X
sgn(σ)
σ∈Σ
n
Y
ai,σ(i)
(D.1.5)
i=1
The following proposition is a useful result for alternating form which also could
have served as a denition.
Proposition D.1.1. Given an alternating form f (x1 , x2 ).
Then:
f (x1 , x2 ) = −f (x2 , x1 ).
(D.1.6)
Proof. Using the linearity of f and that f is alternating gives:
f (x1 , x2 ) = f (x1 , x2 )+f (x1 , x1 ) = f (x1 , x2 +x1 ) = (add and substract f (x2 , x2 +x1 ))
= f (x1 + x2 , x2 + x1 ) − f (x2 , x2 + x1 ) =
= −f (x2 , x2 + x1 ) = −f (x2 , x1 )
Remark D.1.1. Using the property in Proposition D.1.1 gives that an equivalent
denition of an alternating form is that f (x1 , . . . , xn ) = 0 whenever x1 , . . . , xn
is linearly dependent.
The next goal is to dene the operation exterior product on alternating forms.
Before doing this the following function class, n-linear continuous alternating
mappings are needed.
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D.1 Dierential forms
D VECTOR SPACES
Denition D.1.2 (n-linear continuous alternating mappings).
The space of nlinear continuous alternating mappings X n → Y where X, Y are normed vector
spaces is denoted by An (X, Y ).
Let f ∈ An1 (X, Y ), g ∈ An2 (X, Z) together with a bilinear continuous map:
(D.1.7)
φ : Y × Z → V,
where X, Y, Z, V are normed vector spaces. Now create the mapping
h : X n1 +n2 → V by:
h(x1 , . . . , xn1 +n2 ) = φ(f (x1 , . . . , xn1 ), g(xn1 +1 , . . . , xn1 +n2 ))
(D.1.8)
Clearly h is multilinear and continuous but not generally alternating. Instead
it is only alternating when considered as a function of the rst n1 variables or
of the last n2 . Denote the space of such maps An1 ,n2 (X, V ).
Now dene an association between members h ∈ An1 ,n2 (X, V ) and h0 ∈ An1 +n2 (X, V )
by:
ϕn1 ,n2 : An1 ,n2 (X, V ) → An1 +n2 (X, V ),
(D.1.9)
where ϕn1 ,n2 (h) is the multilinear mapping:
X
h0 =
sgn(σ)h(xσ(1) , . . . , xσ(n1 +n2 ) ),
(D.1.10)
σ∈S
where the sum is over the set S of all permutations σ of {1, . . . , n1 + n2 } satisfying:
σ(1) < . . . < σ(n1 ),
(D.1.11)
σ(n1 + 1) < . . . < σ(n1 + n2 ).
(D.1.12)
This mapping is in fact an alternating form which is shown in the following
proposition:
Proposition D.1.2. [Car06] The mapping h0 dened in
nating map.
(D.1.10)
is an alter-
Proof. Divide the permutations satisfying (D.1.11) into two cases and let xi =
xi+1 :
a) Both σ −1 (i) and σ −1 (i + 1) are less than or equal to n1 or that both are
larger or equal to n1 + 1. From equation (D.1.10) and that h is alternating
when considering the rst n1 variables or the last n2 variables, h0 is 0.
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D.1 Dierential forms
D VECTOR SPACES
b) The case left is that σ −1 (i) ≤ n1 and σ −1 (i + 1) ≥ n2 or that σ −1 (i + 1) ≤ n1
and σ −1 (i) ≥ n2 . Consider the permutation τ given by replacing i and i + 1.
Now using τ , pair each permutation satisfying σ −1 (i) ≤ n1 and
σ −1 (i + 1) ≥ n2 with a permutation satisfying σ −1 (i + 1) ≤ n1 and
σ −1 (i) ≥ n2 . This gives rise to pairs which satises when added together:
sgn(σ)h(xσ(1) , . . . , xσ(n1 +n2 ) ) − sgn(σ)h(xτ (σ(1)) , . . . , xτ (σ(n1 +n2 )) ) = 0.
(D.1.13)
The expression (D.1.13) is zero for every pair because xi = xi+1 , thus every
term in the sum (D.1.10) is zero and hence h0 is alternating.
An exterior product is a mapping between two multilinear continuous alternating mappings with respect to a continuous bilinear mapping in the following way:
Denition D.1.3 (Exterior product). Given alternating multilinear forms
f ∈ An1 (X, Y ), g ∈ An2 (X, Y ) in the vector spaces Vf , Vg respectively. The
exterior product ∧ over f, g relative to a bilinear continuous map:
(D.1.14)
φ : Vf × Vg → V,
where V is a vector space is dened as the map:
(D.1.15)
ϕn1 ,n2 (h) ∈ An1 +n2 (X, V )
where h ∈ An1 ,n2 (X, V ) is dened as the element h = (f ∧ g) satisfying:
φ
X
(f ∧ g)(x1 , . . . , xn1 +n2 ) =
sgn(σ) φ(f (xσ(1) , . . . , xσ(n1 ) ), g(xσ(n1 +1) , . . . , xσ(n1 +n2 ) )),
φ
σ∈S
(D.1.16)
where S is the set of permutations satisfying (D.1.11).
Remark D.1.2. The ∧ will be omitted in the notation f ∧ g when Vf = V ,
φ
Vg = R and
(D.1.17)
φ : Vf × Vg → V
is simply multiplication between a vector and a scalar.
Example D.1.2 (Exterior product). Let n1 = 1 in the denition of the exterior
product. Then the exterior product (f ∧ g)(x1 , . . . , xn +1 ) is given by:
φ
(f ∧ g)(x1 , . . . , xn2 +1 ) =
φ
2
nX
2 +1
(−1)i+1 φ(f (xi ), g(x1 , . . . , xi−1 , xi+1 , . . . , xn2 +1 )).
i=1
(D.1.18)
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D.1 Dierential forms
D VECTOR SPACES
Now it is time to dene dierential forms. It is a mapping from an open set to
the space of n-linear continuous alternating mappings.
Denition D.1.4 (Dierential form). Given normed vector spaces X, Y and
an open set U ∈ X . A mapping:
(D.1.19)
f : U → An (X, Y )
is a dierential n-form dened in U with values in Y and f is said to be of class
C k if it is k -times continuously dierentiable.
Remark D.1.3. Using the notation in the denition of a dierential form, a
0-form is dened as a mapping f : U → Y and also that the space of n-forms
(k)
dened in U with values at Y which is of class C k and is denoted as Ωn (U, Y ),
is a vector space.
Example D.1.3 (Dierential form). Given a function f
which is smooth and U is an open set U ∈ R3 . The map:
df =
: U → R, f (x, y, z)
∂f
∂f
∂f
dx +
dy +
dz
∂x
∂y
∂z
(D.1.20)
is a 1-form dened in U with values in A1 (R3 , R) of class C ∞ and is thus in
the space Ω(∞)
(U, R).
1
When dierential forms have been dened, operations on these objects can be
specied. The exterior product for multilinear alternating forms have already
been dened and following the same approach for dierential forms, the exterior
product for dierential forms comes naturally.
Given vector spaces X1 , X2 , X3 and a continuous linear map:
φ : X1 × X2 → X3 .
(D.1.21)
a ∈ Ωkn1 (U, X1 ), b ∈ Ωkn2 (U, X2 )
(D.1.22)
Now consider elements:
where U is an open set in the vector space V . For x ∈ U , the elements a(x) and
b(x) are in the spaces An1 (V, X1 ), An2 (V, X2 ) respectively. Then the exterior
product between these elements is given by:
a(x) ∧ b(x) ∈ An1 +n2 (V, X3 )
(D.1.23)
µ : U → An1 +n2 (V, X3 ),
(D.1.24)
φ
Now the mapping:
124
D.1 Dierential forms
D VECTOR SPACES
where
(D.1.25)
µ(x) = a(x) ∧ b(x),
φ
is of class C k due to a ∈ Ωkn1 (U, X1 ) and b ∈ Ωkn2 (U, X2 ).
Denition D.1.5 (Exterior product for dierential forms).
Given the dierential forms a, b, normed vector spaces X1 , X2 , X3 , V and an open set U ∈ V .
The exterior product of a and b relative to the map
φ : X1 × X2 → X3
(D.1.26)
µ : U → An1 +n2 (V, X3 ),
(D.1.27)
µ(x) = a(x) ∧ b(x).
(D.1.28)
is the dierential form:
where:
φ
Remark D.1.4. If the dierential forms a and b are of class C n , the dierential
form µ in the denition is of class C n . Furthermore if a and b are n1 , n2 -forms
respectively the dierential form µ is a n1 + n2 -form.
a ∧ b(x; ξ1 , . . . , ξn1 +n2 ) is explicitly given by:
φ
X
a ∧ b(x; ξ1 , . . . , ξn1 +n2 ) =
sgn(σ) φ(a(x; ξσ(1) , . . . , ξσ(n1 ) ), b(x; ξσ(n1 +1) , . . . , ξσ(n1 +n2 ) )),
φ
σ∈S
(D.1.29)
using (D.1.16).
Example D.1.4 (Exterior product for dierential forms). Given a function
f : U → X2 , where U ⊂ X1 is an open subset, X1 , X2 , X3 are normed vector
spaces and the n-form µ : U → An (X3 , R). Let the map φ be dened by vector
multiplication between X2 and R. Then the exterior product f ∧ µ is given by:
φ
f ∧ µ(x; ξ1 , . . . , ξn ) =
φ
P
sgn(σ) φ(f (x), µ(x; ξ1 , . . . , ξn ) =
σ∈S
= f (x) · µ(x; ξ1 , . . . , ξn ),
where f · µ is the vector product.
Example D.1.5 (Exterior product for dierential forms). Given dierential
1-forms a, b with scalar values and the map φ : R × R → R which is scalar
125
D.1 Dierential forms
D VECTOR SPACES
multiplication. Then the exterior product a ∧ b(x; ξ1 , ξ2 ) is given by:
a ∧ b(x; ξ1 , ξ2 ) = a(x; ξ1 )b(x; ξ2 ) − a(x; ξ2 )b(x; ξ1 )
(D.1.30)
The next goal is to associate an operation d on µ ∈ Ωkn (U, X) such that
dµ ∈ Ωk−1
n+1 (U, X).
(D.1.31)
µ : U → An (X1 , X)
(D.1.32)
Let the map:
be of class C n , where U ⊂ X1 is an open set. Now considered the dierentiation
of the map µ, which will be a map:
µ0 : U → A1,n (X1 , X)
(D.1.33)
that satises for each x ∈ X1 and i ∈ {1, . . . , n},
(µ0 (x)ξi )(ξ1 , . . . , ξi−1 , ξi+1,...,ξn ) ∈ X.
(D.1.34)
This gives that µ0 is n-linear and an alternating function when considering the
variables ξ1 , . . . , ξi−1 , ξi+1 , . . . , ξn and thus µ0 (x) ∈ A1,n (X1 , X).
The composite of the map (D.1.31) and φ1,n : A1,n (X1 , X) → An+1 (X1 , X)
from Denition D.1.3 is a map dµ satisfying (D.1.31).
Denition D.1.6 (Exterior dierentiation). Given normed vector spaces X1 , X .
The exterior dierential dµ of a dierential form µ is a map which consists of a
composition of two maps:
µ0 : U → A1,n (X1 , X),
(D.1.35)
φ1,n : A1,n (X1 , X) → An+1 (X1 , X).
(D.1.36)
Explicitly dµ(x; ξ1 , . . . , ξn ) = φ1,n (µ0 (x; ξ1 , . . . , ξn )) is given by:
dµ(x; ξ1 , . . . , ξn ) =
n
X
(−1)i+1 (µ0 (x) · ξi ) · (ξ1 , . . . , ξi−1 , ξi+1 , . . . , ξn ). (D.1.37)
i=1
Remark D.1.5. An n-form µ of class C k has an exterior dierential dµ which is
an n + 1-form of class C k−1 .
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D.1 Dierential forms
D VECTOR SPACES
Example D.1.6 (Exterior dierentiation). Given a dierential form µ ∈ Ωk1 (U, X),
where U ∈ X1 and X1 , X are normed vector spaces. Then the exterior dierential of µ is given by:
(D.1.38)
dµ(x; ξ1 , ξ2 ) = (µ0 (x) · ξ1 ) · ξ2 − (µ0 (x) · ξ2 ) · ξ1 .
Example D.1.7 (Exterior dierentiation). Given a function f
the exterior dierential of f is given by:
: U → X.
Then
(D.1.39)
df (x; ξ) = f 0 (x) · ξ.
The following theorems in this section will give a number of usable rules for
calculations regarding dierentials and dierential forms.
The rst theorem shows the result of taking the exterior dierential of a dierential form multiplied with a function. The result resembles the usual "product
rule" when dierentiating the product of two functions.
Theorem D.1.1. [Car06] Given a function
n-form µ. Then:
f
of class C 1 and a dierential
d(f · µ) = (df ) ∧ µ + f · (dµ).
(D.1.40)
Proof. Let f : U → X1 , µ : U → X2 where X1 , X2 , X3 are normed vector
spaces, U ∈ X where U is an open set in the normed vector space X and let
φ : X1 × X2 → X3
(D.1.41)
be a continuous bilinear mapping. Let the mapping:
f · µ : U → An+1 (X1 , X)
(D.1.42)
f · µ(x) = φ(f (x), µ(x)).
(D.1.43)
be given by:
Now dierentiating this function with respect to x at a vector ξ ∈ X gives:
(f · µ)0 (x; ξ1 , . . . , ξn ) = φ(f 0 (x) · ξ, µ(x)) + φ(f (x), µ0 (x) · ξ).
(D.1.44)
Then using (D.1.37) with µ replaced with f · µ gives:
d(f · µ)(x; ξ1 , . . . , ξn ) =
n
X
(−1)i+1 ((f · µ)0 (x) · ξi ) · (ξ1 , . . . , ξi−1 , ξi+1,...,ξn )
i=1
(D.1.45)
127
D.1 Dierential forms
D VECTOR SPACES
Combining (D.1.44) and (D.1.45) together gives:
d(f ·µ)(x; ξ1 , . . . , ξn ) =
n
X
(−1)i+1 φ(f 0 (x)·ξ, µ(x))+
i=1
n
X
(−1)i+1 φ(f (x), µ0 (x)·ξi )
i=1
(D.1.46)
and continuing with this expression (D.1.46) gives (D.1.40) as seen in [Car06].
The next theorem gives the result of taking the exterior dierential of a exterior
product of dierential forms.
Theorem D.1.2. [Car06] Given dierential forms a ∈ Ωkn
Then:
d(a ∧ b) = (da) ∧ b + (−1)n1 a ∧ (db).
1
(U, R), b ∈ Ωkn2 (U, R).
(D.1.47)
Proof. The proof of this theorem is omitted here but can be found in [Car06].
This theorem gives that taking the exterior dierential two times on a dierential forms gives zero.
Theorem D.1.3. [Car06] Given a dierential form
k ≥ 2. Then:
d(dµ) = 0.
µ ∈ Ωkn1 (U, R)
where
(D.1.48)
Proof. The proof of this theorem is omitted here but can be found in [Car06].
The following three theorems will be representation theorems where dierent
dierential forms will be represented in a "canonical" way.
Let X be a nite m-dimensional space. Now choosing a basis for X gives an
identication between X and Rm . Let ui ∈ L(Rm , R) be the i:th coordinate
function where L(Rm ) is the space of linear functions from Rm to R. Given an
open set U ⊂ Rm , let xi be the dierentiable map U → R such that xi is the
restriction of ui to U .
Then the exterior dierential of xi is the constant mapping (i.e every element
in the domain is mapped on a single element) U → L(Rm , R), on the element
ui ∈ L(Rm , R).
Then the rst representation theorem gives the following canonical representation of a dierential form.
128
D.1 Dierential forms
D VECTOR SPACES
Theorem D.1.4. [Car06] Given an open set U ∈ R and a dierential form
Then µ can be represented uniquely in the following canonical
µ ∈ Ωm
n (U, X).
form:
X
µ=
ci1 ,...,in (x)dxi1 ∧ . . . ∧ dxin ,
(D.1.49)
i1 <...<in
where ci1 ,...,in are functions U → X of class C k and i1 , . . . , in are integers
satisfying:
i1 < . . . < in , 1 ≤ i1 , . . . , in ≤ m.
(D.1.50)
Proof. The proof of this theorem is omitted here but can be found in [Car06].
The next representation theorem deals with how the exterior dierential of a
function can be represented canonically.
Theorem D.1.5. [Car06] Given an open set U ⊂ Rm , a normed vector space
X and a function f : U → X of class C 1 . Then the exterior dierential of f
can be represented on the form:
df =
m
X
∂f
dxi .
∂x
i
i=1
(D.1.51)
Proof. The proof of this theorem is omitted here but can be found in [Car06].
Let henceforth in this appendix U be an open set in Rm . The next thing to
consider is the exterior multiplication of two dierential forms canonically represented. Because exterior multiplication is distributive with respect to addition,
let the two dierential n1 , n2 forms a,b respectively only consist of one term:
a = α(x)dxi1 ∧ . . . ∧ dxin1
(D.1.52)
b = β(x)dxj1 ∧ . . . ∧ dxjn2
(D.1.53)
Let the exterior multiplication ∧ be with respect to the bilinear mapping:
φ
φ : X1 × X2 → X,
(D.1.54)
where X1 , X2 are the spaces of values of a, b respectively. Then a ∧ b is given by:
φ
a ∧ b = φ α(x), β(x) dxi1 ∧ . . . ∧ dxin1 ∧ dxj1 ∧ . . . ∧ dxjn2 .
(D.1.55)
φ
In (D.1.55) the factor φ α(x), β(x)
is known and the factor
dxi1 ∧ . . . ∧ dxin1 ∧ dxj1 ∧ . . . ∧ dxjn2
129
(D.1.56)
D.1 Dierential forms
D VECTOR SPACES
can be separated into two cases:
a) When dxi1 , . . . , dxin1 , dxj1 , . . . , dxjn2 are
not distinct.
Then Theorem D.1.3 gives that (D.1.55) is 0.
b) When dxi1 , . . . , dxin1 , dxj1 , . . . , dxjn2 are distinct.
Now choose the permutation σ such that i1 , . . . , in1 , ji , . . . , jn2 are in strictly
increasing order
k1 , k2 , . . . , kn1 +n2
(D.1.57)
and thus
dxi1 ∧ . . . ∧ dxin1 ∧ dxj1 ∧ . . . ∧ dxjn2 = sgn(σ)dxk1 ∧ . . . ∧ dxkn1 +n2 (D.1.58)
Finally what is left is to calculate the exterior dierential of a dierential form
represented canonically. Once more consider a dierential form consisting of
only one term which can be done without loss of generality.
Theorem D.1.6. [Car06] Given a dierential form:
µ = c(x)dxi1 ∧ . . . ∧ dxin ,
(D.1.59)
where c is a mapping U → X of class C 1 . Then:
dµ = dc ∧ dxi1 ∧ . . . ∧ dxin .
(D.1.60)
Proof. If µ consists of dierential forms µ1 , . . . , µn , where µi is an ni -form, such
that
µ = µ1 ∧ . . . ∧ µm .
(D.1.61)
Then utilizing Theorem D.1.2 repeatedly on (D.1.60) gives:
dµ = dµ1 ∧ µ2 ∧ . . . ∧ µm + (−1)n1 µ1 ∧ dµ2 ∧ . . . ∧ µm +
+ . . . + (−1)n1 +...+nm−1 µ1 ∧ . . . ∧ µm−1 ∧ dµm .
(D.1.62)
Now applying (D.1.62) on (D.1.60) gives the following due to c(x)dxi1 = c(x) ∧
130
D.1 Dierential forms
D VECTOR SPACES
dxi1 :
dµ = dc ∧ dxi1 ∧ . . . ∧ dxin + c d(dxi1 ) ∧ dxi2 ∧ . . . ∧ dxin +
+ . . . ± c dxi1 ∧ . . . ∧ dxin −1 ∧ d(dxin ).
(D.1.63)
and using Theorem D.1.3 gives that:
d(dxi1 ) = 0, . . . , d(dxin ) = 0
(D.1.64)
dµ = dc ∧ dxi1 ∧ . . . ∧ dxin .
(D.1.65)
and thus:
131
E FUNCTIONS
E
Functions
In this section some elementary notions about functions will be covered. Example of subject covered are domains, codomains of function, continuously dierentiable functions and bijective functions.
Denition E.0.7 (Domain of function). Given a function f : X → Y . The
domain of f is X (see Figure 15).
Figure 15: Domain and codomain of a function f
Example E.0.8 (Domain of function). Consider the function f
by: f (x) = x2 . The domain of f is R
: R → R,
given
Denition E.0.8 (Codomain of function). Given a function f : X → Y . The
codomain of f is Y (see Figure 15).
Example E.0.9
given by:
(Codomain of function).
Consider the function f : R → R,
f (x) = x2 .
(E.0.66)
The codomain of f is R
Example E.0.10 (Codomain of function). Consider the function f
given by:
f (x) = x2 .
The codomain of f is R+
133
: R → R+ ,
(E.0.67)
E FUNCTIONS
When dealing with expressions concerning partial derivatives, the expressions
easily gets long and complicated so before dening dierentiability, the concept
of multi-indices will be introduced to simplify the problem with partial derivatives.
Denition E.0.9 (Multi-index).
An n-dimensional multi-index α is an n-tuple
α = [α1 , ..., αn ] where α1 , ..., αn ∈ N0 .
The following operations for multi-indices are dened below as properties which
satisfy for all α, β ∈ Nn0 :
Property 1 (Partial ordering).
The partial ordering α ≤ β is equivalent to:
αi ≤ βi ,
(E.0.68)
for all i ∈ {1, ..., n}.
Property 2
(Component-wise addition and subtraction). Addition and subtraction of multi-indices are given by:
α ± β = [α1 ± β1 , ..., αn ± βn ]
(E.0.69)
and for the subtraction to be dened, α ≤ β needs to be satised.
Property 3 (Absolute value).
The absolute value of α is given by:
|α| = α1 + ... + αn .
Property 4 (Factorial).
The factorial of α is given by:
α! = α1 ! · ... · αn !.
Property 5 (Power).
(E.0.70)
(E.0.71)
The power α of x , xα is given by:
αn
1
xα = xα
1 · ... · xn .
(E.0.72)
Example E.0.11 (Multi-index). Given the multi-indices
α = [1, 2, 3, 4, 0, 5]
(E.0.73)
β = [0, 1, 3, 2, 0, 3].
(E.0.74)
and
134
E FUNCTIONS
Here
α + β = [1, 3, 6, 6, 0, 8]
(E.0.75)
and β ≤ α. Furthermore |α| = 15 and α! = 120
Denition E.0.10 (Continuously dierentiable). A function f : Rn → R is k
times continuously dierentiable (class C k ) if all the partial derivatives:
Dα f =
∂ |α| f
,
n
· · · ∂xα
n
1
∂xα
1
|α| ≤ k
(E.0.76)
exist and are continuous for all x = [x1 , .., xn ] ∈ Rn .
Remark E.0.6. A function which is innitely times continuously dierentiable
is of class C ∞ and is called smooth. Furthermore
C ∞ ⊂ C k+1 ⊂ C k ⊂ C 0 .
(E.0.77)
Remark E.0.7. A function f : Rm → Rn , m ≥ n is k-times continuously dierentiable if every component of f = [f1 , . . . , fn ] is k -times continuously dierentiable.
Example E.0.12 (Continuously dierentiable). Given the multi-index
α = [1, 0, 0]. A partial derivative of f at x = [x1 , x2 , x3 ] is
Dα f =
∂f
.
∂x1
(E.0.78)
Example E.0.13 (Continuously dierentiable). Given the multi-index
α = [1, 0, 1]. A partial derivative of f at x = [x1 , x2 , x3 ] is
Dα f =
∂2f
.
∂x1 ∂x3
(E.0.79)
Example E.0.14 (Continuously dierentiable). The function f (x, y) = xy on
belongs to the classes
R2
C 0 , C 1 , ..., C ∞ .
Denition E.0.11 (Bijection). A function f : X → Y is a
following is fullled for all x ∈ X and y ∈ Y .
a) Every element of X is mapped on exactly
one element of Y (Function with domain X )
135
(E.0.80)
bijection if the
E FUNCTIONS
b) Every element of Y was mapped on by at least one element of X (Surjection)
c) No element of Y was mapped on by more than one element of X (Injection)
Example E.0.15 (Bijection). The function f : R → R, given by f (x) = x + 1
is a bijection. Condition a) is satised by choosing x ∈ R in the domain. This
element is mapped on the unique element x+1. To see that b) is satised, choose
y ∈ R in the codomain. This element was mapped on by y − 1 and it is also
unique so c) is also fullled.
Example E.0.16 (Bijection). The function f : R → R, given by f (x) = x2
is not a bijection. Condition a) is satised by choosing x ∈ R in the domain.
This element is mapped on the unique element x2 . To see that b) is not satised, choose for example −1 ∈ R in the codomain. This element was mapped on
by any element in the domain so f is not surjective. Condition c) is also not
fullled, for example choosing 1 ∈ R in the codomain. This element was mapped
on by the two elements −1, 1 ∈ R in the domain so f is not injective.
Example E.0.17 (Bijection). The function f : X → Y where X = {−1, 0, 1},
and given by f (x) = x + 1 is a bijection. Condition a) is satised by choosing x ∈ X in the domain. This element is mapped on the unique
element x + 1. To see that b) is satised, choose y ∈ Y in the codomain. This
element was mapped on by y − 1 and it is also unique so c) is also fullled.
Y = {0, 1, 2}
Example E.0.18
(Bijection). The function f : X → Y where X = {−1, 0, 1},
and given by f (x) = x + 1 is not a bijection. Condition a)
is satised by choosing x ∈ X in the domain. This element is mapped on the
unique element x + 1. To see that b) not is satised, choose −1 ∈ Y in the
codomain. This element was not mapped on by any element in the domain so f
is not surjective. But c) is true, choose y ∈ Y \ {−1}. This element was mapped
on by the unique element y − 1 and the element −1 ∈ Y was not mapped on by
any element so f is injective.
Y = {−1, 0, 1, 2}
136
F EUCLIDEAN GEOMETRY
F
Euclidean geometry
Denition F.0.12 (Euclidean structure). Given a vector space X . An Euclidean structure has the following forms for inner product, norm, angles and
metric for all vectors x, y ∈ X , where x = [x1 , ..., xn ] and y = [y1 , ..., yn ]:
a) hx, yi = x · y =
n
P
xi yi (Inner product)
i=1
b) kxk =
√
x · x (Norm)
c) The angle θ (0 ≤ θ ≤ π) is given by: (Angle)
x·y
−1
θ = cos
kxk · kyk
d) The metric:
d(x, y) = kx − yk (Euclidean metric)
(F.0.81)
(F.0.82)
Example F.0.19 (Euclidean structure). Given the vector space R. Then the
following is an Euclidean structure for all x, y ∈ R.
a) hx, yi = xy (Inner product)
b) kxk = |x| (Norm)
c) θ = cos
−1
xy
|x|·|y|
(Angle)
d) d(x, y) = |x − y| (Metric)
Denition F.0.13 (Euclidean space).
An n-dimensional real coordinate space
together with an Euclidean structure is the n-dimensional Euclidean space which
will be denoted E n .
Example F.0.20 (Euclidean space). R together with the Euclidean structure
in Example F.0.19 is the Euclidean space E .
137
G NOTATION
G
Notation
The notation which will be used is presented below.
x, y, p, points in sets.
Ac , the complement of a set A.
n, m, dimensions in dierent spaces.
γ , a curve or path.
l(γ), the length of the curve γ .
l(γ(a), γ(b)), the length of the curve segment from γ(a) to γ(b).
Ω, a domain.
[a, b], notation for vectors.
a b
, notation for matrices.
c d
AT , the transpose to a matrix A.
R, the real numbers.
R+ , the non-negative real numbers.
R− , the non-positive real numbers.
C, the complex numbers.
Z, the integers.
Q, the rational numbers.
N, the positive integers not including 0.
N0 , the positive integers including 0.
Re(z), is the real part x of z = x + iy .
z , the conjugate of z .
f, g , functions and maps.
f ◦ g , is the composition f of g .
d, a metric or distance.
dI , an intrinsic metric.
139
G NOTATION
Bd (x, r), open ball with center at x, radius r and metric d.
Bd (x, r), closed ball with center at x, radius r and metric d.
B + (x, r), a forward metric ball centered at x with radius r.
X, Y , will usually be used to denote topological, metric, Hausdor, vector , inner product spaces or other spaces.
{Xk }, a sequence X0 , X1 , X2 , . . ..
lim Xk , the limit of the sequence {Xk }.
n→∞
σ , a topology.
U, V, W , neighborhoods of points.
(G, •) or G, a group G with operation •.
(F, +, •) or F , a eld F with operations +, •.
∼, an equivalence relation.
[a]X , the equivalence class of a in X .
L, a linear map.
E , a basis.
X ∗ , the dual space of X .
hx, yi, an inner product.
k·k, a norm.
α, β , multi-indices.
C k , the class of k -times continuously dierentiable functions.
E d , an n-dimensional Euclidean space.
I , an index set.
I, an interval.
I[a,b] , the closed interval between a and b.
I(a,b)] , the open interval between a and b.
Dfp , the dierential map of f at p.
dfp , the dierential of f at p.
140
G NOTATION
p, a point in a manifold.
M, N , for manifolds and submanifolds.
D2 , the 2-dimensional unit disc.
Sn , the n-dimensional sphere.
Cpk (M ), the germs of functions of the class C k on the set M at p.
f ≡ c, the function f : X → Y is for every x ∈ X , f (x) = c where c ∈ R.
Tp M , a tangent space at p for a manifold M .
T M , the tangent bundle for a manifold M .
B , a bilinear form.
f |X = g|X , the functions satises f (x) = g(x) for every x ∈ X .
g , a Riemannian metric.
[X, Y ], the Lie bracket of X, Y .
[X, Y ]p , the Lie bracket of X, Y at a point p.
Dx f , the directional derivative of f at the direction x.
F , a Finsler structure.
SM , the sphere bundle of M .
A, the Cartan tensor in Finsler geometry.
Nji , the non-linear connection.
`, a distinguished section.
ω , a Hilbert form.
Λ, a linear connection.
An (X, Y ), the space of n-linear continuous mapping between X n → Y .
Ln (X, Y ), the space of n-linear mappings between X n → Y .
(k)
Ωn (U, Y ), the space of n-forms dened in U with values at Y and of class C k .
da, the exterior dierentiation of a dierential form a.
141
REFERENCES
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144
Index
Abelian group, 105
Absolute homogeneous, 66
Abstract manifold, 32
Abstract submanifold, 32
Accumulation point, 75, 81
Alternating multilinear form, 117
Arzela-Ascoli theorem, 12
Atlas, 28
Ball, 67, 73
Base, 92
Basis, 111, 125
Bijection, 27, 90, 131
Bilinear form, 43, 119
Binary operator, 103, 110
Binary relation, 103
Bolzano-Weierstrass theorem, 89
Bounded, 73
Bounded sequence, 12
Boundedly compact, 15, 56
Dieomorphism, 28, 67
Dierential form, 121
Dierential map, 25
Dilation, 76
Discrete metric, 71, 85
Discrete topology, 27, 78, 92, 99, 100
Distinguished section, 62
Distributive, 106
Domain, 8, 100
Domain of function, 129
Dual basis, 44, 112
Dual space, 44, 112
Einstein summation convention, 59
Equivalence class, 32, 34, 104
Equivalence relation, 34, 104
Equivalent atlases, 32
Equivalent paths, 3
Euclidean inner product, 133
Euclidean metric, 133
Euclidean norm, 133
Canonical projection map, 61
Euclidean space, 5, 133
Cantor´s diagonalization argument, 90 Euclidean structure, 133
Cartan tensor, 61
Exterior dierentiation, 123
Cartesian product, 35, 103
Exterior product, 120, 122
Cauchy sequence, 13, 74
Cauchy-Schwarz inequality, 114
Field, 106, 110
Chart, 32
Finsler manifold, 57
Chern connection, 64
Finsler structure, 58
Christoel symbols, 52, 61
First variation of arc length, 66
Clopen set, 82
Forward bounded, 69
Closed, 77
Forward boundedly compact, 69
Closed set, 15, 82
Forward Cauchy sequence, 69
Codomain of function, 129
Forward complete, 69
Compact, 14, 67, 84
Forward metric ball, 68
Complete, 19, 56, 75
Fundamental tensor, 61
Connected, 5, 54, 69, 79
Geodesic, 19, 53, 66
Continuous, 27, 76, 80
Germs, 36
Continuous map, 2
Group, 105
Continuously dierentiable, 25, 131
Convex, 5, 101
Hausdor space, 16, 28, 100
Cotangent space, 62
Heine-Borel theorem, 15, 87
Countable, 13, 90
Hermitian matrix, 59, 109
Covariant derivatives, 63
Hilbert form, 62
Covector, 59
Homeomorphism, 27
Curve, 2, 99
Hopf-Rinow-Cohn-Vossen theorem, 20,
56, 69
Dense, 13, 91
Horizontal subspace, 63
Derivation, 39
Determinant, 59, 107, 118
145
INDEX
Inner product, 43, 113
Inner product space, 114
Interior point, 84
Intrinsic metric, 4, 54, 66
Jacobi identity, 48
Koszul formula, 50
Lebesgue number, 98
Lebesgue´s Lemma, 98
Length of curve, 54, 65, 99
Length space, 5, 54, 66
Lie bracket, 47
Limit, 75, 82
Lindelöf compactness, 95
Linear connection, 64
Linear map, 39, 111
Linearly dependent, 118
Linearly independent, 111
Lipschitz constant, 76
Lipschitz continuous, 10, 75, 99
Locally compact, 15, 56
Locally Minkowskian manifold, 58
Lorentzian metric, 45
Maximal atlas, 31
Metric, 55, 66, 71
Metric on Finsler manifold, 66
Metric on Riemannian manifold, 54
Metric space, 14, 71, 72
Minimal geodesic, 20
Minkowski norm, 58
Multi-index, 130
INDEX
Precompact, 22, 68
Pulled-back bundle, 61
Real coordinate space, 71
Rectiable, 99
Rectiable curve, 10
Riemannian connection, 49
Riemannian Finsler manifold, 58
Riemannian manifold, 46, 58
Riemannian metric, 45, 58
Sasaki metric, 63
Semi-metric, 67
Semi-Riemannian metric, 45
Separable, 92
Sequence, 74
Sequentially compact, 17, 89, 96
Sesquilinear, 44
Shortest path, 4, 56, 70
Sphere bundle, 60
Standard topology, 15, 29, 79
Subeld, 106
Submanifolds of Euclidean spaces, 26
Submersion, 25
Surjection, 132
Sylvester´s criterion, 59, 109
t-curve, 65
Tangent bundle, 35, 57
Tangent in abstract manifold, 34, 40
Tangent in submanifold, 33
Tangent space, 58
Tangent space in abstract manifold, 34
Tangent space in submanifold, 33
Tensor, 61, 112
n-linear continuous alternating mappings,Topological space, 2, 36, 77
119
Topology, 55, 68, 78
Natural curve, 6, 53
Triangle inequality, 71, 116
Natural projection, 57
Trivial topology, 80, 100
Neighborhood, 15, 26, 81
Net, 16
u-curve, 65
Non-expanding, 76
Uniform convergence, 7
Non-linear connection, 62
Uniform convergence of curves, 8
Norm, 58, 116
Uniformly continuous, 76
Open set, 73, 78
Parametrization, 3
Path, 2
Path-connected, 79
Permutation, 107, 119
Piecewise smooth variation, 65
Pointwise convergence, 7
Positive-denite, 45, 58, 108
Vector, 59, 110
Vector eld, 36
Vector space, 41, 43, 100, 110
Vertical subspace, 63
146