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Mathematical & Mechanical
Method in Mechanical Engineering
Dr. Wang Xingbo
Fall，2005
Mathematical & Mechanical
Method in Mechanical Engineering
An Introduction to Manifolds
1. Review of Topology
2. Concepts of manifolds
3.
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An Introduction to Manifolds
How can we describe it?
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An Introduction to Manifolds
• Many engineering objects have a shape of
complicated surface. These complicated surfaces
can be described by manifolds. Theories of
manifolds have exhibited their elegances and
excellences in many aspects of engineering, e.g.
in controlling of robots, in structural analysis of
mechanical engineering. Manifold is regarded
to be a powerful tool for a senior engineer or a
researcher to master.
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Review of Topology
A topological space is a pair (X, T)
where X is a set and T is a class of
subsets of X, called topology, which
satisfies the following three properties.
(i) X, ∈T.
(ii) If { X i}i∈I∈T, then ∪i∈I X i ∈T
(iii) If X 1,…, X n ∈T, then ∩i=1,…,n X i∈T.
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Review of Topology
If (X, T) is a topological space, the elements
of T are said open sets.
A subset K of X is said closed if its
complementary set X \K is open
The closure Uof a set U X is the intersection of

all the closed sets K X with U K

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Closure of a set
K1
K2
X
K4
U
K3
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Continuous function
If (X,T) and (Y,U) are topological spaces,
a mapping f : X→Y is said continuous if is
open for each T∈U
f
(X,T )
T
f 1
(Y,U )
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Homeomorphism
An injective, surjective and continuous
mapping f : X→Y, whose inverse mapping
is also continuous, is said homeomorphism
from X to Y. If there is a homeomorphism
from X to Y these topological spaces are
said homeomorphic.
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Base, second countable
If (X,T) is a topological space, a class
B Tis said base of the topology, if each
open set turns out to be union of elements of
B. A topological space which admits a countable
base of its topology is said second countable.
If (X,T) is second countable, from any base B
it is possible to extract a subbase B’ B 
which is countable.
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Topology generated by set-class
If A is a class of subsets of X≠; and CA is
the class of topologies T on X with AT,
TA := T∈CA T is said the topology generated
by A. Notice that CA≠ because the set
of parts of X, P(X), is a topology and
includes A.
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topology induced on a set
If AX, where (X,T) is a topological space, the pair
(A,TA) where, TA := {UA | UT},defines a topology
on A which is said the topology induced on A by X.
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Neighborhood
•
If (X,T) is a topological space and pX, a
neighborhood of p is an open set UX with
p∈U. If X and Y are topological spaces and
xX, f: X→Y is said to be continuous in X,
if for every neighborhood of f(x), VY ,
there is a neighborhood of x, UX, such
that f(U) V . It is simply proven that f :
X→Y as above is continuous if and only if
it is continuous in every point of X.
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Connect
• A topological space (X,T) is said
connected if there are no open sets A,
B≠ with AB = and AB = X. It
turns out that if f: X→Y is
continuous and the topological space
X is connected, then f(Y) is a
connected topological space when
equipped with the topology induced by
the topological space Y.
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Hausdorff
•
A topological space (X,T) is said Hausdorff if each
pair (p,q)XX admits a pair of neighborhoods Up,
Uq with p∈Up, q∈Uq and UpUq=. If X is
Hausdorff and xX is a limit of the sequence
{Xn}n∈NX, this limit is unique.
Up
p
Uq
q
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Semi-distance
A semi metric space is a set X endowed with a semidistance.
d: XX→[0,+∞], with
d ( x, y )  d ( y , x )
d ( x , y )  d ( y , z )  d ( x, z )
d ( x, y )  0  x  y
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Open Ball
•
The semidistance is called distance and the semi
metric space is called metric space.
•
An open metric balls are defined as
Bs ( y ) : {z  R | d ( z, y )  s}
n
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Connected by path
•
A topological space (X,T) is said connected
by paths if, for each pair p, qX there is
a continuous path : [0,1] →X such that(0)
= p,(1) = q,
q=(1)

p=(0)
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Cover
If X is any set, a covering of X is a class {Xi}i∈I,
XiX for all iI, such that i∈I Xi = X
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Compactness-Finite Cover
A topological space (X,T) is said compact if from each
covering of X, {Xi}i∈I are made of open sets, it is possible to
extract a covering {Xj}j∈I of X with j finite. This is also
called a finite covering property
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Group
Let G be a set and  be a operation defined on W. If W and
 satisfy the following regulations:
1. There is a unit e in G such that g  G, e  g = g
2. . g  G, g 1  G  g  g 1  e | g 1  g  e
3.
g1  G , g 2  G  kg1  g 21  G where
Then G is called a group over R
kR
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isomorphism
Let S and T be tow groups with operations ,  respectively.
If there exists a one-to-one mapping : S T such that,
for any s1 , s2  S , t1 , t2  T
1
1

t


(
s

s
)
1. Ift   ( s ), t   ( s )
it results tin
1
1
2
2
1
2. If
1
2
2
et , esare unit in S and T respectively, then
et   (es )
then S is said to be isomorphic to T, or vice versa;
the mapping  is said to be a isomorphism between S and T.
Two isomorphic groups can be regarded to have the same
structure algebraically
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Concepts of Manifolds
A topological space (X, T) is said topological manifold of
dimension n if X is Hausdorff, second countable and is
locally homeomorphic to Rn, i.e., for every pX there is
a neighborhood pUp and a homomorphism p: Up→Vp
where VpRn is a open set.
Up
p
p
VpRn
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Chart
(n-chart)Let X be topological space, U is an
open subset of X. Let  be a homeomorphism
from U X to an open subset V Rn, namely,
: p→(x1(p),…,xn(p)). Then the ordered
pair (U, )= C is called an n-chart on M.
where Rn is the n-dimensional Euclidean
space.
A chart can be thought of a mapping from some
open set to an open subset of Rn
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Chart
C=( U,)
U

VRn
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K-Compatible
•
Let (U, ) and (U, ) be two charts on
a topological space M . If UU, let V
and V be image of UU under corresponding
homeomorphisms  and . The two charts
are said to be compatible if  -1 viewed
as a mapping from V Rn to V Rn, is a C
function. If UU= then the charts are
also said to be compatible. If  -1 and
 -1 are all Ck (k<) functions, then
 and  are said to Ck-compatible. If any
 and  are said to C-compatible, then M
is said to be smooth.
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k-Compatible
M
U
U


-1
V Rn
V Rn
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Atlas
An atlas A on a topological space M is a
collection of charts{C} on M such that
1. Any two charts in atlas are piecewise
k-compatible;
2. A covers M, i.e. M  C A U
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Differential structure, Differentian Manifolds
A differential structure on a topological
space is an atlas with the property that any
chart that is compatible with the charts of
the atlas is also an element of the atlas.
An n-dimensional differential manifold M is a
topological space endowed with a differential
structure of n-charts.
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Local Cooridnate Systems
If M is a n-dimensional differential manifold,
then any point PM has such a open
neighborhood U that is homeomorphism to an
open set V of Rn, or we can say, that there
exists at least one open subset U of M that
has a n-chart (U,) such that (P)=VRn. At
this time, the coordinate ((P))iof image(P)
corresponding to P is called coordinate of
PU and is denoted by xi(P)=((P))i. (U, xi)is
called a local coordinate system.
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Local Cooridnate Systems
It can be seen that, two charts (U,), (V,) on an ndimensional differential manifold M are related two local
coordinate systems. If UV，then there also exist two
local coordinate systems corresponding to UV. Thus any
point P UV has two coordinate representations
xi(P)=((P))i and yi(P)=( (P))i and the two are dependent.
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Local Cooridnate Systems
P =  - 1  (P)   - 1  (P)
xi (P)  ( (P))i  ( ( 1 ( (P)))i  (  1 )( (P))i  (  1 )( y( P))i
yi (P)  ( (P))i  ( ( 1 ( (P)))i  (  1 )( (P))i  (  1 )( x( P))i
   f ,   g
1
1
f  g 1 , g  f 1
xi ( P )  f ( y i ( P )), y i ( P )  g ( xi ( P ))
i  1, 2,..., n
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Differentiable Partitions of Unity on Manifolds
P =  - 1  (P)   - 1  (P)
xi (P)  ( (P))i  ( ( 1 ( (P)))i  (  1 )( (P))i  (  1 )( y( P))i
yi (P)  ( (P))i  ( ( 1 ( (P)))i  (  1 )( (P))i  (  1 )( x( P))i
   f ,   g
1
1
f  g 1 , g  f 1
xi ( P )  f ( y i ( P )), y i ( P )  g ( xi ( P ))
i  1, 2,..., n
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Tensor Fields in Manifolds and Associated
Geometric Structures
Tangent , Tangent Bundle and State Space .
Local representative
Let (t ) be a continuous function from R to a differential
manifold M.

VRn
(t)
p
C=(U,)
R
-

  
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Tangent , Tangent Bundle and State Space
. Local representative

VRn
(t)
p
C=(U,)
R
-

  
 
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Related
Two curves f and g are said to be related at p
if and only if
1. f(0)=g(0)=p;
2.The
derivatives
of
the
representations of f and g are equal
dx if
dt
|t 0 
dxgi
dt
|t 0
local
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Related properties
If f(t) and g(t) are related in chart
(U, ), they are also related in
chart (V, )
1
 g    

1
f    
 g

f
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Related properties
1
d

d

d
d
d

1

 ( g (t )) 
 (  (  ( g (t )))) 
  ( g (t ))
1
dt
dt
d   d   dt
1
d

d 
d
d
d

1
 ( f (t )) 
 (  (  ( f (t )))) 
  ( f (t ))
1
dt
dt
d   d   dt
f(t) and g(t) are related in chart (U, )
d
d
 ( f (t )) |t 0   ( g (t )) |t 0
dt
dt
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Related properties
1
1
d

d

d 
d 
d
d


  ( f (t )) |t 0 
  ( g (t )) |t 0
1
1
d   d   dt
d   d   dt
1
1
D(     )  
d  d  
 ( 1
)
d  d 
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Related Properties
1
D(  
)   ( g (0)) 
d
d
  ( g (t )) |t 0  D (    1 )   ( f (0))    ( f (t )) |t 0
dt
dt
d
d
  ( f (t )) |t 0    ( g (t )) |t 0
dt
dt
Vp
Up p
f(t)
g(t)
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Tangent space
If M is a differentiable manifold and pM, the tangent
space at point p, denoted as TpM, is defined to be the
set of all equivalent classes Qp at p in M.
TpM has the same dimension as M
Define a map
is injective
d
d  (p)([ ])  (   ) |t 0
dt
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Tangent space
For any v in Rn, choose  such that for any |t|<  (p)  tv
is a path through  (p)
in  (U ) and 1 (p  tv )
is a smooth path through p
1
d  (p)([ (p  tv)])  v
d  (p)
is bijective , a linear isomorphic map from
TpM to Rn
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Basis induced by a chart
Let M be a differentiable manifold, pM, and take a chart
(U,) with pU. If E1,…,En is the canonical basis of Rn,
then e pi   p Ei (i  1, 2,..., n)
define a basis in TpM which we call the basis induced in
TpM by the chart (U,)
(U,), (V, ) with pU,V and induced basis on
p
Tp M
p
{
e
{ei }i 1,2,...,n j } j 1,2,...,n t p  t ieip  t j e jp Tp M
x
t  k | ( p ) t k
x
j
j
x
p
e  k | ( p ) e j
x
j
p
k
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Derivations
Symbol D(M) indicates the real vector space of all differential
functions from manifold M to R

|
Dp (M ) indicate the vector space spanned by
k p
x
Let M be a differentiable manifold. A derivation in TpM is
a R-linear map Dp: D(M)→R, such that, for each pair f,
g D(M):
Dp ( fg )  f ( p) D p g  g ( p) D p f
Symbol DpM is used to indicate the R-vector space of the
derivations in p
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Derivation
Let M be a differential manifold. Take any TpM
and any DpDpM
(1) If hD(M) vanishes in a open neighborhood
of p or, more strongly, h = 0 in the whole
manifold M,then
Dph= 0
(2) For every f, gD(M),
Dpf = Dpg
provide f(q) = g(q) in an open neighborhood of p.
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Flander's Lemma
If f: B→R is C∞(B) where BRn is an open
1
2
n
p

(
x
,
x
,...,
x
starshaped neighborhood of 0
0
0
0)
, then there are n differentiable mappings gi:
B→R such that, if p  ( x1 , x 2 ,..., x,n ) then
n
f (p)  f (p 0 )   gi (p)( x i  x0i )
i 1
f
gi (p 0 )  i |p
x 0
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Flander’s lemma
p  ( x1 , x 2 ,..., x n )
y i (t )  x0i  t ( x i  x0i )
d
f (p 0  t (p  p 0 ))dt
0 dt
1 n
f (p 0  t (p  p 0 ) i
i
 f (p 0 )   
(
x

x
0 ) dt
i
0
x
i 1
f (p)  f (p 0 )  
1
f
gi (p)   i |p0 t (p p0 ) dt
0 x
1
f
g i ( p 0 )  i | p0
x
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basis of TpM
Let M be a differentiable manifold and pM. There exists a
R-value vector space isomorphism F: TpM  DpM such
p
that, if {ei }i 1,2,...,n is the basis of TpM induced by any local
coordinate system about p with coordinates (x1,..., xn), it holds:
k
F :t e
t p  t e Tp M
i
p
i
is a basis of DpM
p
k

t
|
k p
x
k
And in particular the set
{

| }
k p k 1,2,..., n
x
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Tangent Bundle
The tangent bundle of a manifold M, denoted by
TM is defined as the union of the tangent
spaces for all pM. That is:
TM   TpM
pM
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Tangent Bundle
TM is itself a differential manifold of dimension 2n
TM= {(p, v) |pM , vTpM}
Tangent bundle is called a state space
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Tangent Bundle
Given two manifolds A and B and a function f:AB,
there is a natural way to form a mapping, denoted by Tf,
from TA to TB
TA
Tf

TB

f
A
B
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Cotangent Space and Phase Space
Let M be an n-dimensional manifold. For each
pM, the dual space Tp*M is called the cotangent
space on p and its elements are called cotangent vectors or differential 1-forms on p.
If (x1,..., xn) are coordinates about p
inducing the basis , the associated dual basis
in Tp*M is denoted by {dxk|p}k=1,…,n.
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Cotangent Space and Phase Space
The cotangent bundle of a manifold M, denoted by T*M is
defined as the union of the cotangent spaces for all pM.
That is:
TM  TM
*
pM
*
p
A cotangent space is also called a phase space that is a
collection of all possible positions and momenta that cab be
obtained by a configuration space.
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Covariant Derivative and Levi-Civita's Connection
Let M be a differentiable manifold. An affine connection or
covariant derivative r, is a map
 : (Y,X )
Y X
where X , Y , Y X are differentiable contravariant vector
fields on M, which obeys the following requirements:
(1)
fY +gZX = f▽Y X + gZ X, for all differentiable
functions f , g and differentiable vector fields X , Y ,Z ;
(2)
YfX = Y(f)X+fY X for all differentiable vector field
X , Y and differentiable functions f ,
(3)
X (Y +Z) = X Y +X Z for all ,R and
differentiable vector fields X, Y, Z.
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Covariant Derivative and Levi-Civita's Connection
In components referred to any local coordinate system
j



Y

j
i j
i
XY   i  Y
XY 
X
j
j
i
j
X

x

x

x
i
i x
x
x


j
if i , j are fixed
define a differentiable tensor
i x
x

field which is the derivative of
with respect to 
j
x
and thus
x i


x i



k
k 
  
, dx  k : ij k
j
j
x
x
x
x
x i
Mathematical & Mechanical
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Covariant Derivative and Levi-Civita's Connection
The coefficients  =  ( p) are differentiable
functions of the considered coordinates and are
called connection coefficients.
k
ij
k
ij
Using these coefficients and the above expansion,
in components, the covariant derivative of Y with
respect to X can be written down as:.
i

X
(Y X)i : X j ( j  ijk Y k )
x
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Covariant Derivative and Levi-Civita's Connection
X i
i
k
i
i


Y

:

X

:
X
, j.
jk
j
j
x
X is called covariant derivative of X (with
respect to the affine connection ). In
components we have
(Y X)i = YjXi,j.
k
h
k
p
q

x

x

x

x

x
k
h
ij  h i j  h i
 pq
j
x x x x x x
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Covariant Derivative and Levi-Civita's Connection
T :   
i
jk
i
jk
i
kj
define a tensor field is represented by

j
k
T ()  (   ) i  dx  dx
x
i
jk
i
kj
This tensor field is symmetric in the covariant indices and is
called torsion tensor field of the connection.
Mathematical & Mechanical
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Covariant Derivative and Levi-Civita's Connection
The assignment of an affine connection on a differentiable
manifold M is completely equivalent to the assignment of

coefficients  ( p)  
in each local
| , dx | 
x
coordinate system, which differentially depend
on the point p and transform as
k
ij

x
k
|
i p
j
p
p
k
h
k
p
q

x

x

x

x

x
h
ijk ( p)  h | p i j | p  h | p
|

pq ( p)
i
j p
x
x x
x
x x
under change of local coordinates.
Mathematical & Mechanical
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Covariant Derivative and Levi-Civita's Connection
If M is endowed with a metric, then the manifold is called a
Riemann manifold. The connection on a Riemann manifold
is called Levi-Civita's affine connection.
Let M be a Riemann manifold with metric locally
represented by. There is exactly one affine
connection  such that :
(1). It is metric, i.e.,  = 0
(2). It is torsion free, i.e., T() = 0.
Mathematical & Mechanical
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Covariant Derivative and Levi-Civita's Connection
That is the Levi-Civita connection which is
defined by the connection coefficients, called
Christoffel's coefficients
1 is g ks g sj g jk
i
i
 jk  { j k } : g ( j  k  s ).
2
x
x
x
Consider a (pseudo) Euclidean space
En. Fixing an orthogonal Cartesian coordinate
system, we can define an affine connection locally
given by:
.
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Covariant Derivative and Levi-Civita's Connection
Y 
 X Y : X
j
j
x x
i
j
Mathematical & Mechanical
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Meaning of the Covariant Derivative
X(p + hY )  X(p)
l im
h 0
h
Unfortunately, there are two problems involved in
the formula above:
(1) What does it mean p+hY ? In general, we have
not an affine structure on M and we cannot move
points thorough M under the action of vectors as
in affine spaces. (The reader should pay attention
on the fact that affine connections and affine
structures are different objects!).
Mathematical & Mechanical
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Meaning of the Covariant Derivative
X(p) TpM but X(p + hY ) Tp+hYM. If something
like p + hY makes sense, we expect that p + hY ≠
p because derivatives in p should investigate the
behavior of the function qX(q) in a
“infinitesimal” neighborhood of p. So the
difference X(p + hY ) - X(p) does not make sense
because the vectors belong to different vector
spaces! .
Mathematical & Mechanical
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Meaning of the Covariant Derivative
Let M be a differentiable manifold equipped with an affine
connection . If X and Y are differentiable contravariant
vector fields in M and pM
 Y X(p)  lim
h 0
P1[p,  (h)]X( ( h))  X(p)
h
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where  : [0 , ] → M is the unique geodesic
segment referred to r starting from p with
initial tangent vector Y(p) and
P[(u),(v)]:T(u)T(v)
is the vector-space isomorphism induced by the r parallel
transport along a (sufficiently short) differentiable curve
: [a , b] → M for u < v and u , v[a , b].