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IOSR Journal of Mathematics (IOSR-JM)
e-ISSN: 2278-5728, p-ISSN: 2319-765X. Volume 12, Issue 2 Ver. I (Mar. - Apr. 2016), PP 121-136
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Applications Geometry Riemannian Manifolds
Mohamed M. Osman
1
(Department Of Mathematics Faculty Of Science - University Of Al-Baha – K.S.A)
Abstract: In This Paper Some Fundamental Theorems , Definitions In Riemannian Geometry Manifolds In The
Space R To Pervious Of Differentiable Manifolds Which Are Used In An Essential Way In Basic Concepts Of
Applications Riemannian Geometry Examples Of The Problem Of Differentially Projection Mapping
Parameterization System By Strutting Rank k ..
n
Keywords : Basic Notions On Differential Geometry – Tangents Spaces And Vector Fields – Differential
Geometry – Cotangent Space And Vector Bundles – Tensor Fields – Differentiable Manifolds Charts - Surface
N-Dimensional.
I.
Introduction
A Riemannian Manifolds Is A Generalization Of Curves And Surfaces To Higher Dimension , It Is
Euclidean In E In That Every Point Has A Neighbored, Called A Chart Homeomorphism To An Open Subset
Of R , The Coordinates On A Chart Allow One To Carry Out Computations As Though In A Euclidean Space
, So That Many Concepts From R , Such As Differentiability, Point Derivations , Tangents , Cotangents
Spaces , And Differential Forms Carry Over To A Manifold. In This We Given The Basic Definitions And
Properties Of A Smooth Manifold And Smooth Maps Between Manifolds , Initially The Only Way We Have To
Verify That A Space , We Describe A Set Of Sufficient Conditions Under Which A Quotient Topological Space
Becomes A Manifold Is Exhibit A Collection Of C Compatible Charts Covering The Space Becomes A
Manifold , Giving Us A Second Way To Construct Manifolds , A Topological Manifolds C Analytic
Manifolds , Stating With Topological Manifolds , Which Are Hausdorff Second Countable Is Locally Euclidean
Space , We Introduce The Concept Of Maximal C Atlas , Which Makes A Topological Manifold Into A
Smooth Manifold , A Topological Manifold Is A Hausdorff , Second Countable Is Local Euclidean Of
Dimension n , If Every Point p In M Has A Neighborhood U Such That There Is
A
Homeomorphism  From U Onto A Open Subset Of R , We Call The Pair A Coordinate Map Or Coordinate
System On U , We Said Chart ( U ,  ) Is Centered At p  U ,  ( p )  0 , And We Define The Smooth Maps
f : M  N Where  M , N  Are Differential Manifolds We Will Say That f Is Smooth If There Are
Atlases ( U  , h ) On M And ( V  , g  ) On N . In This Paper, The Notion Of A Differential Manifold Is
n
n
n



n
Necessary For The Methods Of Differential Calculus To Spaces More General Than De R n , A Differential
Structure On A Manifolds M Induces A Differential Structure On Every Open Subset Of M , In Particular
Writing The Entries Of An  n  k  Matrix In Succession Identifies The Set Of All Matrices With R n , k , An
n  k Matrix Of Rank k Can Be Viewed As A K-Frame That Is Set Of k Linearly Independent Vectors In
R
n
, V
n ,k
K  n  Is
Called The Steels Manifold ,The General Linear Group
GL ( n )
By The Foregoing
V n ,k
Is Differential Structure On The Group n Of Orthogonal Matrices, We Define The Smooth Maps Function
f : M  N Where M , N Are Differential Manifolds We Will Say That f Is Smooth If There Are Atlases
,  V , g  On N , Such That The Maps  g f h   Are Smooth Wherever They Are
Defined f Is A Homeomorphism If Is Smooth And A Smooth Invers .
A Differentiable Structures Is
Topological Is A Manifold It An Open Covering U  Where Each Set U  Is Homeomorphism, Via Some
U 
, h
 On
Homeomorphism
In
1
M
B
h
To An Open Subset Of Euclidean Space
Consists Of An Open Subset
M
B
B
U  M
R
n
, Let
And A Homeomorphism
h
M
Be A Topological Space , A Chart
Of
U
Onto An Open Subset Of R m ,
A C Atlas On M Is A Collection  U  , h  Of Charts Such That The U  Cover M And h , h  The
Differentiable Vector Fields On A Differentiable Manifold M , Let X And Y Be A Differentiable Vector
Field On A Differentiable Manifolds M Then There Exists A Unique Vector Field Z Such That Such That ,
For All f  D , Zf  ( XY  YX ) f If That p  M And Let x :U  M Be A Parameterization At Specs .
1
r
B
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Applications Geometry Riemannian Manifolds
II.
A Basic Notions On Differential Geometry
In This Section Is Review Of Basic Notions On Differential Geometry:
2.1 First Principles
Hausdrff Topological 2.1.1
A Topological Space M Is Called (Hausdorff) If For All x , y  M There Exist Open Sets Such That
x  U And y  V And U  V  
Definition 2.1.2
A Topological Space M Is Second Countable If There Exists A Countable Basis For The Topology
On M .
Definition 2.1.3: Locally Euclidean Of Dimension ( M )
A Topological Space M Is Locally Euclidean Of Dimension N If For Every Point
On Open Set U  M And Open Set w  R So That U And W Are (Homeomorphism).
xM
There Exists
n
Definition 2.1.3
A Topological Manifold Of Dimension N Is A Topological Space That Is Hausdorff, Second Countable
And Locally Euclidean Of Dimension N.
Definition 2.1.4
A Smooth Atlas
Of A Topological Space
A
Is Given By: (I) An Open Covering  U  Where
M
i I
Ui  M
Open And
That If U
i
M   i I U i
U
j
 
.(Ii) A Family  
Then The Map  U
:U
i
i
i
Definition 2.1.5
If U  U    Then The Diffoemorphism
i
j
i
U
 Wi
j
 
 i U i  U
j
j
i I Of Homeomorphism  i Onto Open Subsets
U  U  Is ( A Diffoemorphism )
i
Wi  R
n
So
j
   U
j
i
U
j
 Is Known As The (Transition Map).
Definition 2.1.6
A Smooth Structure On A Hausdorff Topological Space Is An Equivalence Class Of Atlases, With
Two Atlases A And B Being Equivalent If For U ,    A And V ,    B With U  V   Then The
i
Transition
 i U i  V j    j U i  V j  Map
Definition 2.1.7
A Smooth Manifold
Smooth Structure
M
 F 
1
Definition 2.1.9
A Map F : M 
F : N  M Is Also Smooth.
i
j
Of Dimension N Is A Topological Manifold Of Dimension N Together With A
:  (U )   (V )
p M
j
n
Definition 2.1.8
Let M And N Be Two Manifolds Of Dimension
Smooth At p  M If There Exist Charts U ,   , V ,   With
The Composition 
It Smooth At Every
j
i
Is A Diffoemorphism (As A Map Between Open Sets Of R ).
Respectively A Map F : M  N Is Called
And F ( p )  V  N With F (U )  V And
m,n
p U  M
Is A Smooth ( As Map Between Open Sets Of
R
n
Is Called Smooth If
.
N
Is Called A Diffeomorphism If It Is Smooth Objective And Inverse
1
Definition 2.1.10
A Map F Is Called An Embedding If
Definition 2.1.11
If F : M
 N
F
Is An Embedding Then
Is An Immersion And (Homeomorphism) Onto Its Image.
F (M )
Is An Immersed (Sub Manifolds) Of
N
.
2.2 Tangent Space And Vector Fields
Let C ( M , N ) Be Smooth Maps From M And N , Let C ( M ) Smooth Functions On M Is Given A
Point p  M Denote, C ( p ) Is Functions Defined On Some Open Neighbourhood Of p And Smooth At p .



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Applications Geometry Riemannian Manifolds
Definition 2.2.1
(I) The Tangent Vector
The Formula
To The Curve
X
c :   ,    M
 d ( f  c) 
X ( f )  c (0 ) ( f )  

dt


(1)
(Ii) A Tangent Vector
 (0 )  p
This Is
At
X
At
: f  C
t0
Is The Tangent Vector At
pM

X   ( 0 ) : C ( p )  R
Is The Map
t  0

c (0 ) :C

( c (0 ) )  R
Given By
c (0 )
t 0
Of Some Curve
 :   ,    M
With
.
Remark 2.2.2
A Tangent Vector At p Is Known As A Liner Function Defined On
(Leibniz Property)
(2)
.
X ( f g )  X ( f )g  f X (g ) , f , g  C ( p)
C

( p)
Which Satisfies The

2.3 Differential Geometrics
Given F  C ( M , N ) And


,
p M
 Choose A Curve
X  TpM
 :( ,  )  M
With
 (0 )  p
And
 ( 0 )  X
This Is Possible Due To The Theorem About Existence Of Solutions Of Liner First Order Odes ,
Then Consider The Map F : T M  T N Mapping X  F ( X )  ( F   ) ( 0 ) , This Is Liner Map Between Two
/
* p
p
*p
F ( p)

Vector Spaces And It Is Independent Of The Choice Of
Definition 2.3.1
The Liner Map
Image
F* p ( X )
F* p
.
Defined Above Is Called The Derivative Or Differential Of
Is Called The Push Forward
At
X
pM
F
At
While The
p
.
Definition 2.3.2: Cotangent Space And Vector Bundles And Tensor Fields
Let M Be A Smooth N-Manifolds And p  M .We Define Cotangent Space At p Denoted By
T M To Be The Dual Space Of The Tangent Space At p : T ( M )   f : T M  R  , f Smooth Element Of
*
*
p
p
*
Tp M
Are Called Cotangent Vectors Or Tangent Convectors At
T p M  T f ( p)R  R
*
Vector At
p
Is Called
df
p
p
p
.(I) For
And Referred To The Differential Of
(Ii) For A Chart U
, , x
i
 Of
M
And
p U
f :M  R
.Not That
f
Then  dx 
i
n
i 1
df
Smooth The Composition
 Tp M
*
p
Is A Basis Of
*
Tp M
So It Is A Cotangent
In Fact  dx  Is The
i
n
Dual Basis Of
 d 

i 
 dx  i  1
.
Definition 2.3.3
A Smooth Real Vector Bundle Of Rank k Denoted  E , M ,   Is A Smooth Manifold E Of Dimension
n  1 The
Total Space A Smooth Manifold M Of Dimension n The Manifold Dimension n  k And A Smooth
Subjective Map  : E  M (Projection Map) With The Following Properties: (I) There Exists An Open Cover
 V    I Of
1
 :  (V )  V  R .(Ii)
Diffoemorphism
For
p  M ,   ( p )    p   R  R And We Get A Commutative Diagram ( In This Case
Projection Onto The First Component .(Iii) Whenever V  V   The Diffoemorphism.
M
And
1

k
k


k



(3)
Takes The Form
1
   
1

 
: V   V 
 p , a    p , A  ( p ) ( a )  , a 
R
k
Any
Point
Is
 1 : V  R  V
k


Where
R
k
 V   V   R
k
A  :V   V   GL ( k , R )
Is Called Transition Maps.
Definition 2.3.4 : Bundle Maps And Isomorphism’s
~ ~
Suppose  E , M ,   And E , M , ~  Are Two Vector Bundles A Smooth Map F
~
:E  E
Is Called A
 . (I) There Exists A Smooth Map
Smooth Bundle Map From  E , M ,   To 
Such That
The Following Diagram Commutes That   F ( q )   f  ( q )  For All p  M (Ii) F Induces A Linear Map From
~
f :M  M
~ ~ ~
E , M ,
E
p
To
~
E f ( p)
For Any
p M
.
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Applications Geometry Riemannian Manifolds
Definition 2.3.5 : Projective Spaces
The n  Dimensional Real (Complex) Projective Space, Denoted By P
n
n 1
(R)
or
Pn ( C ))
, Is Defined As
n 1
The Set Of
1-Dimensional Linear Subspace Of R
or C
) , P ( R ) or P ( C ) Is A Topological Manifold.
Definition 2.3.6
For Any Positive Integer n , The n  Torus Is The Product Space T  ( S  ...  S ) .It Is A
n  Dimensional Topological Manifold. (The 2-Torus Is Usually Called Simply The Torus).
Definition2.3.7
The Boundary Of A Line Segment Is The Two End Points; The Boundary Of A Disc Is A Circle. In
General The Boundary Of A n  Manifold Is A Manifold Of Dimension ( n  1) , We Denote The Boundary Of A
Manifold M As  M . The Boundary Of Boundary Is Always Empty,  M  
Lemma 2.3.8
Every Topological Manifold Has A Countable Basis Of Compact Coordinate Balls. Every
Topological Manifold Is Locally Compact.
Definitions 2.3.9
Let M Be A Topological Space n -Manifold. If (U ,  ), (V , )  Are Two Charts Such That U  V    ,
n
n
n
1
1
The Composite Map    :  (U  V )    (U  V )  Is Called The Transition Map From  To  .
Definition 2.3.10
An Atlas A Is Called A Smooth Atlas If Any Two Charts In A Are Smoothly Compatible With Each
Other. A Smooth Atlas A On A Topological Manifold M Is Maximal If It Is Not Contained In Any Strictly
Larger Smooth Atlas. (This Just Means That Any Chart That Is Smoothly Compatible With Every Chart In A Is
Already In A.
Definition 2.3.11
A Smooth Structure On A Topological Manifold M Is Maximal Smooth Atlas. (Smooth Structure Are
Also Called Differentiable Structure Or C Structure By Some Authors).
Definition 2.3.12
A Smooth Manifold Is A Pair ( M , A), Where M Is A Topological Manifold And A Is Smooth Structure
On M . When The Smooth Structure Is Understood, We Omit Mention Of It And Just Say M Is A Smooth
Manifold.
Definition 2.3.13
Let M Be A Topological Manifold.
(I) Every Smooth Atlases For M Is Contained In A Unique Maximal Smooth Atlas. (Ii) Two Smooth Atlases
For M Determine The Same Maximal Smooth Atlas If And Only If Their Union Is Smooth Atlas.
Definition 2.3.14
Every Smooth Manifold Has A Countable Basis Of Pre-Compact Smooth Coordinate Balls. For
Example The General Linear Group The General Linear Group GL ( n , R ) Is The Set Of Invertible n  n -Matrices
With Real Entries. It Is A Smooth n -Dimensional Manifold Because It Is An Open Subset Of The n Dimensional Vector Space M ( n , R ) , Namely The Set Where The (Continuous) Determinant Function Is
Nonzero.
Definition 2.3.15
Let M Be A Smooth Manifold And Let p Be A Point Of M . A Linear Map X : C ( M )  R Is Called A
Derivation At p If It Satisfies:
(4)
X ( fg )  f ( p ) Xg  g ( p ) Xf
For All f , g  C ( M ) . The Set Of All Derivation Of C ( M ) At p Is Vector Space Called The Tangent Space
To M At p , And Is Denoted By [ T M ]. An Element Of T M Is Called A Tangent Vector At p .
1

2
2



p
p
Lemma 2.3.16
Let M Be A Smooth Manifold, And Suppose
pM
Then Xf  0 . If f ( p )  g ( p )  0 , Then X ( fp )  0 .
Definition2.3.17
If  Is A Smooth Curve (A Continuous Map 
Manifold M , We Define The Tangent Vector To  At
d
dt
|t

Is The Standard Coordinate Basis For T
d
 

(t ) 
 (t  ) ,
dt


 d
And 
 dt

|t  t 


t
R
And X
:J  M
t  J
 TpM
, Where
. If
Is A Const
f
J  R
And Function,
Is An Interval) In A Smooth
To Be The Vector   ( t

 d

)   
|t   T  (t ) M
 dt



, Where
. Other Common Notations For The Tangent Vector To  Are
. This Tangent Vector Acts On Functions By:
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Applications Geometry Riemannian Manifolds

  (t  ) f   
(5)

d

|t  f 
|t
dt

d


dt

f


 d( f )

 (t  )
dt


.
Lemma 2.3.18
Let M Be A Smooth Manifold And p  M . Every X  T M  Is The Tangent Vector To Some Smooth
Curve In M .
Definition 2.3.19
A Lie Group Is A Smooth Manifold G That Is Also A Group In The Algebraic Sense, With The
Property That The Multiplication Map m : G  G   G And Inversion Map m : G  G  , Given
By m ( g , h )  g h , i ( g )  g , Are Both Smooth. If G Is A Smooth Manifold With Group Structure Such That
The Map G  G  G Given By ( g , h )  gh Is Smooth, Then G Is A Lie Group. Each Of The Following
Manifolds Is A Lie Group With Indicated Group Operation. The General Linear Group GL ( n , R ) Is The Set Of
Invertible  n  n  Matrices With Real Entries. It Is A Group Under Matrix Multiplication, And It Is An Open
Sub-Manifold Of The Vector Space M ( n , R ) , Multiplication Is Smooth Because The Matrix Entries Of A Aid B .
p
1
1
1
Inversion Is Smooth Because Cramer’s Rule Expresses The Entries Of A As Rational Functions Of The Entries
Of A . The n  Torus T  ( S  ...  S ) Is A n  Dimensional A Belgian Group.
Definition 2.3.20 Lie Brackets
Let V And W Be Smooth Vector Fields On A Smooth Manifold M . Given A Smooth
Function f : M  R , We Can Apply V To f And Obtain Another Smooth Function Vf , And We Can Apply
f  W V f ,
W To This Function, And Obtain Yet Another Smooth Function W V  f  W (Vf ) . The Operation
However, Does Not In General Satisfy The Product Rule And Thus Cannot Be A Vector Field, As The
n
1
1
  
V  

 x 
Following For Example Shows Let
And
  

W  


 y 
On R , And Let
n
f ( x, y )  x, g ( x, y)  y
. Then Direct
Computation Shows That V W ( f g )  1 , While  f V W g  g V W f   0 , So V W Is Not A Derivation Of C ( R ) .
We Can Also Apply The Same Two Vector Fields In The Opposite Order, Obtaining A (Usually Different)
Function W V f . Applying Both Of This Operators To f And Subtraction, We Obtain An


2

Operator [V , W ] : C ( M )  C ( M ) ,
Called
The
Lie
Bracket
Of
Defined
V And W ,
By [V , W ] f   V W  f  WV  f . This Operation Is A Vector Field. The Smooth Of Vector Field Is Lie Bracket
Of Any Pair Of Smooth Vector Fields Is A Smooth Vector Field.
Lemma 2.3.21: Properties Of The Lie Bracket
The Lie Bracket Satisfies The Following Identities For All V , W , X  ( M ) . Linearity:  a , b  R ,
 [ aV  bW , X ]  a [V , X ]  b [W , X ]

 [ X , aV  bW ]  a [ X , V ]  b [ X , W ].
(6)
(I)
Ant
For
(7)
f,g C
Symmetry [V , W ] 

(M )
 [W , V ]
.(Ii)Jacobi



Identity [V , [
W , X ] ]  [ W , [ X , V ] ]  [ X , [V , W ] ]  0
.
:
[ f V , g W ]  f g [ V , W ]  ( f V g ) W  ( g W
f) V

2.4 Convector Fields
Let V Be A Finite – Dimensional Vector Space Over R And Let V Denote Its Dual Space. Then V
Is The Space Whose Elements Are Linear Functions From V To R, We Shall Call Them Convectors. If   V
Then  : V  R For The Any v  V , We Denote The Value Of  On v By  v  Or By v ,  . Addition And
*
*
*
Multiplication
By
Scalar
In
Are
Defined
By
The
V
Equations     v    v    v  ,   v     v   . Where v  V ,  ,   V And   R .
Proposition 2.4.1 : Convectors
Let V Be A Finite- Dimensional Vector Space. If ( E ,..., E ) Is Any Basis For V ,Then The Convectors
*

1
2
1
2
1
(  ,...,  )
1
n
n
Defined By:
 (E j )  
(8)
Form A Basis For
i
V

i
j
1
 
0
,Called The Dual Basis To ( E
DOI: 10.9790/5728-1221121136
j
)
i  j
if
i  j
if
.Therefore,
dim V

 dim V
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Applications Geometry Riemannian Manifolds
Definition 2.4.2 Convectors On Manifolds
AC  Convector Field  On M , r  0 , Is A Function Which Assigns To Each   M A Convector
  T  M  In Such A Manner That For Any Coordinate Neighborhood U ,   With Coordinate Frames E ,.., E ,
r

p
P
1
n
The Functions   E  , i  1,....., n , Are Of Class C On U . For Convenience, "Convector Field” Will Mean
C  Convector Field.
Remark 2.4.3
It Is Important To Note That A C  Convector Field  Defines A Map  :   M   C  M  , Which Is Not Only
r
i

r
r
– Linear But Even C  M   Linear, More Precisely, If f , g  C  M  Any X , Y Are Vector Fields On M ,
Then   f X  g Y   f   X   g  Y  . For These Functions Are Equal At Each p  M .
Definition 2.4.4: Tensors Vector Spaces
We Now Proceed To Define Tensors. Let k  N Given A Vector Space V ,....., V  One Can Define A
Vector Space V  .....  V  Called Their Tensor Product. The Element Of This Vector Space Are Called Tensors
With The Situation Where The Vector Space V ,....., V Are All Equal To The Same Space. In Fact The Tensor
r
R
r
1
1
k
k
1
k
T V We Define Below Corresponds To  V  .....  V  In The General Notation. And We Define
V  V  ....  V  Be The Cartesian Product Of k Copies Of V .A Map  From V To A Vector Space U Is Called
Multiline If In Each Variable Separately I.E. (With The Other Variables Held Fixed) .
Definition 2.4.5
Let V  V  .....  V  Be The Cartesian Product Of k Copies Of V . A Map  From V To A Vector
Space U Is Called Multiline If It Is Linear In Each Variable Separately ( I.E. With The Other Variables Held
Fixed )
Definition 2.4.6
A (Covariant) K-Tensor On V Is A Multiline Map T : V  R . The Set Of K-Tensors On V Is
Denoted T (V ) . In Particular, A 1-Tensor Is A Linear Form, T (V )  V . It Is Convenient To Add The
Convention That T (V )  R . The Set T (V ) Is Called Tensor Space, It Is A Vector Space Because Sums And
Scalar Products Of Multiline Maps Are Again Multiline.
2.5Alternating Tensors
Let V Be A Real Vector Space. In The Preceding Section The Tensor Spaces T V Were Defined ,
Together With The Tensor Product ( S , T )  S  T , T ( V )  T (V )  T (V ) There Is An Important Construction
Of Vector Spaces Which Resemble Tensor Powers Of V , But For Which There Is A More Refined Structure,
These Are The So-Called Exterior Powers V , Which Play An Important Role In Differential Geometry Because
The Theory Of Differential Forms Is Built On Them. They Are Also Of Importance In Algebraic Topology And
Many Other Fields. A Multiline Map  :V  V  ....  V  U Where k  1 Is Said To Be Alternating If For All
v ,......, v Are Inter-Changed That Is  ( v ,......, v ,...., v )    ( v ,....., v ,....., v ,...., v ) Since Every Permutation Of
Space
*
k
*
1
k
k
k
K
k
k
k
1
0
*
k
k
k
k l
l
k
1
1
k
Numbers
1,......, k
i
k
i
j
i
Can Be Decomposed Into Transpositions, It Follows That
k
 ( v  1 ,...., v  k )  sgn  ( v 1 ,....., v k )
For
Of The Numbers 1,....., k  .For Example Let V  R The Vector Product
( v , v )  v  v  V Is Alternating For V  V   V .And Let V  R The  n  n  Determinant Is Multiyear And
Alternating In Its Columns, Hence It Can Be Viewed As An Alternating Map ( R )  R .
Definition 2.5.1
An Alternating K-Form Is An Alternating K-Tensor V  R The Space Of These Is Denoted A (V ) , It
Is A Linear Subspace Of T (V )
Theorem 2.5.2
Assume Dim V   n With e 1 ,...., e n  A Basis. Let  ,....,    V Denote The Dual Basis . The Elements
  Sk
All Permutations
1
1
1
3
2
n
n
k
k
k
*
1

1
 ....  
i ,k
 Where
Proof:
Let T
Denote By
e
1

j
Is An Arbitrary Sequence Of K Numbers In 1,...., n  ,Form A Basis For
k
T (V )
.
 . Notice That If J  ( j ,....., j ) Is Another Sequence Of K Integers, And We
The Element e ,...., e   V Then T ( e )   That Is T ( e )  1 If J  I And 0 Otherwise. If
 
i1
 ......  
ik
1
k
k
j
j1
Follows That They
Then a
I  ( i1 ,...., i k )
n
 T (e j )  0
T
I
jk
I
j
I j
I
j
Are Linearly Independent, For If A Liner Combination
T    a I T I 
 I

Is Zero,
. It Follows From The Multilinearity That A K-Tensor Is Uniquely Determined By Its Values
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Applications Geometry Riemannian Manifolds
On All Elements In
 T ( e I )T I
Agrees With
V
k
Of The Form e . For Any Given K-Tensor
On All
T
T
j
e
Hence
j
 T ( e I )T I
I
We Have That The K-Tensor
And We Conclude That The
Span T
TI
K
.
(V )
I
2.6 The Wedge Product
In Analogy With The Tensor Product ( S , T )  S  T Form T (V )  T (V )  T (V ) , There Is A
Construction Of A Product A (V )  A (V )  A Since Tensor Products Of Alternating Tensors Are Not
Alternating, It Does Not Suffice Just To Take S  T .
Definition 2.6.1
Let
The
Wedge
Product
Defined
By
 S  T   A (V ) Is
S  A (V ) And T  A (V ) .
 S  T   ALt ( S  T ) .Notice That In The Case k  0 ,Where A (V )  R , The Wedge Product Is Just Scalar
Multiplication.
Example 2.6.2
Let  ,   A (V )  V Then By Definition      1 / 2 (        ) Since The Operator. Alt Is
Linear The Wedge Product Depends Linearly On The Factors S And T. It Is More Cumbersome To Verify The
Associative Rule For  . In Order To Do This We Need The Following.
Lemma 2.6.3
Let R  A (V ) , S  A (V ) And T  A (V ) Then ( R  S )  T  R  ( S  T )  Alt ( R  S  T )
(9)
R  ( S  T )  Alt ( R  Alt ( S  T )  Alt ( R  S  T )
The Wedge Product Is Associative, We Can Write Any Product T  .....  T  Of Tensor T  A (V ) Without
Specifying Brackets. In Fact It Follows By Induction From That T  .....  T   Alt ( T  ......  T ) Regardless Of
How Brackets Are Inserted In The Wedge Product In Particular, It Follows From
k
k
k 1
l
k
k 1
l
k 1
l
k
1
1
*
2
1
k
l
2
1
2
2
1
m
k i
1
1
 1
 .....   k ( v 1 ,...., v k ) 
Basic Elements

1
k!

det  i ( v j ) i , j
 For All v
1
,...., v k   V
Are Written In This Fashion As
I

I
And 

 
r
1
1
,....,  2   V
 .....  
iI
i
r
ik
*
r
Are Viewed As 1-Forms, The
 Where
I  ( i1 ,...., i k )
Is An Increasing
Sequence Form 1,...., n  This Will Be Our Notation For  From Now On. The Wedge Product Is Not
Commutative. Instead, It Satisfies The Following Relation For Interchange Of Factors. In This Defined A
Tensor  On V Is By Definition A Multiline V Denoting The Dual Space To V , r Its Covariant Order And s Its
Contra Variant Order , Assume  r  0  or  s  0  Thus  Assigns To Each R-Tape Of Elements Of V And s Tupelo
I
*
Of Elements Of
The  Fixed The  k
(10)
*
V
A Real Number And If For Each k , 1  k   r  s  We Hold Every Variable Except
The Linearity Condition
 th  Satisfies
 v 1 , .... ,  v k    v k , ..   v 1 , ..., v k     ( v 1 , ....., v k )
For All  ,     R And v , v   V Or V Respectively For A Fixed r , s We Let f (V ) Be The Collection Of All
Tensors On V Of Covariant Order s And Contra Variant Order r , We Know That As A Function
From  V  ....  V  V  ...  V  To Order R They May Be Added And Multiplied By Scalars Elements R With This
Addition And Scalar Multiplication f (V ) Is A Vector Space So That If  ,    f (V ) And
 ,    R Then       Defined In The Way Alluded To Above That Is By.
r
k
s
k
*
r
r
s
1
2
1
1
2
1
2
s
2
      v , v ,....     v , v ,...     v , v ,... 
(11)
Is Multiline And Therefore In f (V ) This f (V ) Has A Natural Vector Space Structure. In Properties Come
Naturally Interims Of The Metric Defined Those Spaces Are Known Interims Differential Geometry As
Riemannian Manifolds A Convector Tensor On A Vector V Is Simply A Real Valued  ( v , v ,...., v ) Of Several
Vector Variables ( v , .... , v ) Of V The Multiline Number Of Variables Is Called The Order Of The Tensor , A
Tensor Field  Of Order r On Linear In Each On A Manifold M Is An Assignment To Each Point p  M Of
Tensor  On The Vector Space T M Which Satisfies A Suitable Regularity Condition C , C Or C As P On
M .
Definition2.6.4
With The Natural Definitions Of Addition And Multiplication By Elements Of R The Set f (V ) Of All
Tensors Of Order r , s On V Forms A Vector Space Of Dimension n .
Definition2.6.5
We Shall Say That   f (V ) , V A Vector Space Is Symmetric If For Each 1  i , j  r ,We
 v , v ,..., v ,..., v ,..., v  Similarly
If Interchanging The i  th And j  th Variables 1  i , j  r Changes The
1
1
2
2
1
r
2
1
1
1
2
2
2
1
2
r
s
s
1
1
2
r
r
0
p

r
p
r
s
rs
r
s
1
2
j
i
r
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Sign,   v
, v 2 ,..., v j ,..., v i ,..., v r
1
 Then We Say  Is Skew Or Anti Symmetric Or Alternating Covariant Tensors Are
Often Called Exterior Forms, A Tensor Field Is Symmetric Respective Alternating If It Has This Property At
Each Point.
Theorem 2.6.8
The Product  f (V )  f (V )    f (V )  Just Defined Is Bilinear Associative If w , ...., w  Is
r
rs
s
1
n
Abasis1 V  f (V ) Then w  ,....,  w  And 1  i ,...., i   n Is A Basis Of f (V ) Finally F : W  V Is
Linear, Then
Proof:
Each Statement Is Proved By Straightforward Computation To Say That Bilinear Means That  ,  Are
*
1
i (1 )
r
i(r )
1
Numbers 
1
, 2   f
r
(V )
And 
Then 
 f (V )
r
 
1
2
*
r

   1       2  

Similarly For The Second
Variable This Is Checked By Evaluating Side On r  s Vectors Of V In Fact Basis Vectors Suffice Because Of
Linearity Associatively Is Similarly             , The Defined In Natural Way This Allows Us To
Drop The Parentheses To Both ( w  ,....,  w ) From A Basis It Is Sufficient To Note That If e ,...., e  Is
i (1 )
i(r )
1
The
Basis
Of
Exactly ( w  ,....,  w
i (1 )
To ( w  ....  w ) Then
The
V Dual
) This Follows From The Two Definitions.
1
i(r )

(12)
(13)
(w
i (1 )
e
( i 1 , . . . ,i r )
 ,...,  w
j (1 )

i(r )
Tensor
n
) (e
,....., e
j (1 )
j(r)
,..., e
j(r)
 
)  w
( i 1 ,...,i r )
n
Defined
Is
if ( i1 ,..., i r )  ( j 1 ,...., j r )

 0

1



Previously 
if ( i1 ,..., i r )  ( j 1 ,....., j r )
i (1 )
(e
j (1 )
)w
i(2)
(e
j(2)
),.., w
i(r )
  
Which Show That Both Tensors Have The Same Values On Any Order Set Of
Are Thus Equal Finally Given F : W  V If  w ,...., w  Then
r
i (1 )
j (1 )
,.., 
i(r )
j(r)

Basis Vectors And
*
rs
1





(14)
F

*


 F

*
  F

*

w rs
( w 1 ),......, F
 F 
*
  w 1 ,....,

F
(w r ) 
*
w
*
*
    F ( w 1 ),......, F ( w r  s )  
1
*
( w 1 ),...., F
*
(w rs )






,...., w r  s
Which Proves F (   )  ( F  )  ( F  ) And Completes Tensor Field.
Remark 2.6.9
The Rule For Differentiating The Wedge Product Of A P-Form  And Q-Form  Is
*
*
*
p

d 
(2.8)
Definition 2.6.10
Let f : M
p

Be A C Map Of
 N

  q  d
C

  q  (  1) 
p
q
 d q
p
p

Manifolds, Then Each C Covariant Tensor Field  On

M By
Tensor
Field F  On
The
Formula
( F  ) ( X ,...., X )  
( F X ,......, F X ) The Map F : f ( N )  f ( M ) So Defined Is Linear And Takes
Symmetry Alternating Tensor To Symmetric Alternating Tensors.
Lemma 2.6.11
Let   0 Be An Alternating Covariant Tensor V Of Order N=Dim. V And Let e , .... , e  Be A Basis Of
N
Determines
A C Covariant
*
*
*
p
1p
rP
F ( p)
*
*
1p
r
r
r p
1
V
Then For Any Set Of Vectors v
Example 2.6.12
(I) Possible P-Forms

p
,..., v n
1
 With v
 i ej
j
i
 ( v 1 , ...., v n )  det 
We Have,
n
j
i
.
In Two Dimensional Space Are.






(15)
The Exterior Derivative Of Line Element
Area d u ( x , y ) dx  v ( x , y ) dy    v   u  dx  dy .
x
(Ii) The Three Space P-Forms

(16)
0
1
2


 u ( x , y ) dx  v ( x , y ) dy 

  ( x , y ) dx  dy


f (x, y)
Givens
The
Two
Dimensional
Curl
Times
The
y
p
Are.







0
 f (x)
1
 v 1 dx
1
 v 2 dx
2
 w 1 dx
3
  ( x ) dx
2
 dx
1
3
 dx
2
 v 3 dx
 w 2 dx
2
 dx
3
3
 dx
1
3
 w 3 dx
1
 dx
2







We See That
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Applications Geometry Riemannian Manifolds

d

d

(17)
i
Where
jk


1


2
i j k
 jvk

1

i j m
2
 1w1  
2
dx
 dx
1
 dx
w2   3w3
m
 dx
1
1
 dx
3





Is The Totally Anti-Symmetric Tensor In 3-Dimensions.The Isomorphism Vectors Tensor Field We
Saw In The Equation


~
V  g (V ,  )  g (  , V )
Vector Spaces Is Provided By
g
And
g

~
~
1
1
V  g (V , )  g ( , V ) The Link Between The Vector And Dual


~
~



A  B Components A  B Then A  B Components B   g  B So
And
1
If


Where A   g   A And B   g   B So Why Do We Bother One-Forms When Vector Are Sufficient The
Answer Is That Tensors May By Function Of Both One-Form And Vectors , There Is Also An Isomorphism A
Mongo Tensors Of Different Rank , We Have Just Argued That The Tensor Space Of Rank ( 1.0) Vectors And
(0.1) Are Isomorphic , In Fact All 2   Tensor Space Of Rank ( m  n ) With Fixed ( m  n ) Are Isomorphic, The
mn
Metric Tensor Like Together These Spaces As Exempla Field By Equation
T 


 g (e  ,T
k


ek )
We Could Now
Use The Inverse Metric





(18)
T   g
( e , T  e ) g
T
 g
g  T
The Isomorphism Of Different Tensor Space Allows Us To Introduce A Notation That Unifies Them, We Could
Affect Such A Unification By Discarding Basis Vectors And One-Forms Only With Components, In General
Isomorphism Tensor Vector A Defined By.
1
k
k
k
k
(19)
And

A  A e
 A g
A  A e

p

e
Transforms Like A Basis One-Form.
2.7: Tensor Fields
The Introduced Definitions Allows One To Introduce The Tensor Algebra
And
R
(T p M )
k



e  A e 

Is Invariant Under A Change Of Basis Because
Obtained By Tensor Products Of Space
p
k
,
(T
*
p
M )
A R (T p M )
Of Tensor Spaces
. Using Tensor Defined On Each Point
p  M One May Define Tensor Fields.
Definition 2.7.1
Let M Be A N-Dimensional Manifold. A Differentiable Tensor Field T Is An Assignment
p  t Where Tensors t  A ( T M ) Are Of The Same Kind And Have Differentiable Components With
p
p
Respect
 

K
 x
To
p
All
R
The

 k  1 ,..., n  T p M

p
Canonical
And
dx
Bases
Of
k  1,..., n   T p M
k
Given
By
Product
Of
Induced By All Of Local Coordinate System
*
p
A R (T p M )
Bases
M
.In
Particular A Differentiable Vector Field And A Differentiable 1-Form
( Equivalently Called
Coveter Field ) Are Assignments Of Tangent Vectors And 1-Forms Respectively As Stated Above.
For Tensor Fields The Same Terminology Referred To Tensor Is Used .For Instance, A Tensor Field t Which Is
Represented In Local Coordinates By
i
t j ( p)

x

 dx
i
j

p
Is Said To Of Order (1,1) .It Is Clear That To
p
Assign On A Differentiable Manifold M A Differentiable Tensor Field T ( Of Any Kind And Order ) It
Necessary
And
Sufficient
To
Assign
A
Set
Of
Differentiable
Functions
.
 x ,...., x   T
 x ,...., x  . In Every Local Coordinate Patch (Of The Whole Differentiable
Structure M Or, More Simply, Of An Atlas Of M ) Such That They Satisfy The Usual Rule Of Transformation
Of Comports Of Tensors Of Tensors If  x ,...., x  And  y ,...., y  Are The Coordinates Of The Same Point
p  M In Two Different Local Charts.
1
n
i 1 ,. .. ., i m
1
n
j 1 ,. .. , j k
1
(20)
T
i 1 , . . . ,i m
j 1 ,...., j k
 

i1

 x




p
n
1
n



 ....  

im

x



 dx
j1

p
 ... 
dx
jk

p
p
(I) It Is Obvious That The Differentiability Requirement Of The Comports Of A Tensor Field Can Be Choked
Using The Bases Induced By A Single Atlas Of Local Charts. It Is Not Necessary To Consider All The Charts
Of The Differentiable Structure Of The Manifold.
(Ii) If X Is A Differentiable Vector Field On A Differentiable Manifold, M Defines A Derivation At Each
Point
p  M : if
f  D M
,
X
( f )  X ( p)
i
p

x
Where
i
x
1
,...., x
n
 Are Coordinates Defined About p . More
p
Generally Every Differentiable Vector Field X Defines A Linear Mapping From D ( M ) To D ( M ) Given By
f  X ( f ) For Everywhere X ( f )  D ( M ) Is Defined As X ( f ) ( P )  X ( f ) For Every p  M .(Iii) For (Contra
p
Variant) Vector Field
X
On A Differentiable Manifold M , A Requirement Equivalent To The Differentiability
DOI: 10.9790/5728-1221121136
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Applications Geometry Riemannian Manifolds
Is The Following The Function X ( f ) : P
Of
f  D (M )
X ( f ):P  X
p
 X
(f)
p
, (Where We Use
X
p
As A Derivation) Is Differentiable For All
. Indeed It X Is A Differentiable Contra Variant Vector Field And If f  D ( M ) , One Has That
( f ) Is A Differentiable Function Too As Having A Coordinate Representation.
X
(21)
( f ) 
1
 :  (U
)
x
1
,..., x
n

i
X
x
1
,...., x
n

f
x
1
i
( x ,. .. ,x
n
)
In Every Local Coordinate Chart (U ,  ) And All The Involved Function Being Differentiable.
Conversely p  X ( f ) Defines A Function In D ( M ) , X ( f ) For Every f  D ( M ) The Components Of
p
p  X
p
(f)

(U ,  )
In Every Local Chart
 .Where The Function
Must Be Differentiable. This Is Because In A Neighborhood Of q  U ,
Vanishes Outside U And Is Defined As r  x ( r ) , h ( r ) In
Where
Is
The
Its
Component
Of
(The
Coordinate

x
x ) And h A Hat Function Centered On q With
U
Support In U . Similarly The Differentiability Of A Covariant Vector Field w Is Equivalent To The
Differentiability Of Each Function p  X . w For All Differentiable Vector Fields X .(Iv) If f  D ( M ) The
X (q )  X f
i
(1 )
f
(1 )
 D (M )
i
i
i
p
Differential Of
In
f
,
p
df
p
p
Is The 1-Form Defined By
df
f

p
x
 dx 
i
i
p
In Local Coordinates About
p
. The
p
Definition Does Not Depend On The Chosen Coordinates .As A Consequence, The Point p  M ,
p  df Defines A Covariant Differentiable Vector Field Denoted By df And Called The Differential Of f . (V)
The Set Of Contra Variant Differentiable Vector Fields On Any Differentiable Manifold M Defines A Vector
Space With Field Given By R Is Replaced By D ( M ) , The Obtained Algebraic Structure Is Not A Vector Space
Because D ( M ) Is A Commutative Ring With Multiplicative And Addictive Unit Elements But Fails To Be A
Field. However The Incoming Algebraic Structure Given By A Vector Space With The Field Replaced By A
Commutative Ring With Multiplicative And Addictive Unit Elements Is Well Known And It Is Called Module.
A Sub Manifolds Of Others Of R For Instance S Is Sub Manifolds Of R It Can Be Obtained As The
Image Of Map Into R Or As The Level Set Of Function With Domain R We Shall Examine Both Methods
Below First To Develop The Basic Concepts Of The Theory Of Riemannian Sub Manifolds And Then To Use
These Concepts To Derive A Equantitive Interpretation Of Curvature Tensor , Some Basic Definitions And
Terminology Concerning Sub Manifolds, We Define A Tensor Field Called The Second Fundamental Form
Which Measures The Way A Sub Manifold Curves With The Ambient Manifold , For Example X Be A Sub
Manifold Of Y Of  : E  X And g : E  Y Be Two Vector Brindled And Assume That E Is Compressible ,
p
2
n
3
3
3
1
p 1
Let f : E  Y And g : E  Y Be Two Tubular Neighborhoods Of X In Y Then There Exists A C .
2.8 : Differentiable Manifolds And Tangent Space
In This Section Is Defined Tangent Space To Level Surface  Be A Curve Is In
R ,  : t   ( t ),  ( t ),....,  ( t )  A Curve Can Be Described As Vector Valued Function Converse A Vector
1
n
1
2
n
Valued Function Given Curve , The Tangent Line At The Point
 d
(t )  

dt
 dt
d
1
t 0 ....,
d

t0 


n
dt
We Many
k
Bout
Smooth Curves That Is Curves With All Continuous Higher Derivatives Cons The Level
Surface f  x , x ,..., x   c Of A Differentiable Function f Where x To i  th Coordinate The Gradient Vector Of
1
f
At Point P
Derivative
2
n

i
 x ( P ), x ( P ),...., x ( P )
1
2
n
 Is
f 
 f
f  
,.....,

1
n
x 
 x
f
 f
1
D u f   f u  
u  ... 
u
1
n

x

x

Given By Equation
f
x
1
(P )
(x
1



n
, The Point
P
f
 (x
 ( P )  .... 
 x )
1
Is Given A Vector
x
n
(P)
On Level Surface
n
 x )
n
f
u  ( u ,..., u )
1
x
1
2
, x ,..., x
n
n
The Direction
 The Tangent Is
 ( P )  0 . For The Geometric Views The
Tangent Space Shout Consist Of All Tangent To Smooth Curves The Point P , Assume That Is Curve Through
t  t Is The Level Surface
f  x , x ,..., x   c That Is f  ( t ),  ( t ),....,  ( t )   c By Taking Derivatives On
1
2
n
1
2
n
0
 f
Both 
 x
1

 f

 x
 P  (   ( t 0 )   ....  
n
(P)
n

( t ))   0

And So The Tangent Line Of


Is Really Normal Orthogonal To
f
Where  Runs Over All Possible Curves On The Level Surface Through The Point P .The Surface M Be
A C Manifold Of Dimension n With k  1 The Most Intuitive To Define Tangent Vectors Is To Use Curves ,

Any Point On
With  ( M )  p Unfortunately It
p  M
DOI: 10.9790/5728-1221121136
And Let  :   ,    M Be A C Curve Passing Through p That Is
Is Not Embedded In Any R The Derivative  ( M ) Does Not Make Sense
1
Be
M
M
N
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Applications Geometry Riemannian Manifolds
,However For Any Chart U ,   At p The Map     At A C Curve In R And Tangent Vector v    v  ( M ) Is Will
Defined The Trouble Is That Different Curves The Same v Given A Smooth Mapping f : N  M We Can Define
How Tangent Vectors In T N Are Mapped To Tangent Vectors In T M With U ,   Choose
1
/
n
p
q
Charts q  f ( p ) For p  N And V ,  For q  M We Define The Tangent Map Or Flash-Forward Of f As A
Given Tangent Vector X     T N And d f : T M , f      f     . A Tangent Vector At A Point p In A
p
p
*
p
*
Manifold M Is A Derivation At p
, Just As For R The Tangent At Point p Form A Vector Space
p
, We Also Write T ( M ) A Differential Of Map f : N  M Be
T ( M ) Called The Tangent Space Of M At
n
p
p

A C Map Between Two Manifolds At Each Point p  N The Map F Induce A Linear Map Of Tangent Space
Called Its Differential At p , F : T N  T N As Follows It X  T N Then Is The Tangent Vector In
*
TF ( p)M
Defined  F
*
(X
p

) f  X
p
f
p
F ( p)
 F  R
p
f C
,

p

. The Tangent Vectors Given Any C -Manifold M Of
(M )
Dimension n With For Any p  M ,Tangent Vector To M At p Is Any Equivalence Class Of C -Curves
Through p On M Modulo The Equivalence Relation Defined In The Set Of All Tangent Vectors At p Is Denoted
By T M We Will Show That Is A Vector Space Of Dimension n Of M .The Tangent Space T M Is Defined As
1
p
p
The Vector Space Spanned By The Tangents At
M
, And The Cotangent T
Space
*
p
Of A Manifold At
M
Differential Line Elements
2.8. : Definition
Let M And
M
1
2
e  dx
i
i
To All Curves Passing Through Point
p  M
  
Ei  

i
 x 
, We Take The Basis Vectors
TpM
p
p
In The Manifold
Is Defined As The Dual Vector Space To The Tangent
For
And We Write The Basis Vectors
TpM
Thus The Inner Product Is Given By
Be Differentiable Manifolds A Mapping
 /  x , dx
 : M
 M
1
 
i
2
j
i
As The
*
Tp M
.
 Is A Differentiable If It Is
1
Differentiable , Objective And Its Inverse  Is Diffoemorphism If It Is Differentiable  Is Said To Be A Local
Diffoemorphism At p  M If There Exist Neighborhoods U Of p And V Of  ( p ) Such That  : U  V  Is A
Diffoemorphism , The Notion Of Diffeomorphism Is The Natural Idea Of Equivalence Between Differentiable
Manifolds , Its An Immediate Consequence Of The Chain Rule That If  :  M  M  Is A Diffoemorphism
Then d  : T M    T M  . Is An Isomorphism For All  :  M  M  In Particular , The Dimensions Of
1
p
M 1 And M
 ( p)
1
2
1
2
d  : T p M 1  T ( p ) M
Are Equal A Local Converse To This Fact Is The Following
2
2
2
Is An Isomorphism
Then  Is A Local Diffoemorphism At p From An Immediate Application Of Inverse Function In R , For
Example Be Given A Manifold Structure Again A Mapping f : M  N In This Case The Manifolds
N And M Are Said To Be Homeomorphism , Using Charts ( U ,  ) And ( V , ) For N And M Respectively We
n
1
~
Can Give A Coordinate Expression f : M  N
Definition 2.8.2
Let M  And M  Be Differentiable Manifolds And Let  :  M  M  Be Differentiable Mapping
For Every p  M And For Each v  T M  Choose A Differentiable Curve  : (   ,  )  M With  ( M )  p And
1
1
1
2
1
p
1
 ( 0 )  v


Take
And  : M
1
1
 
The
1
1
d
Mapping
p
: T ( p ) M
2
By
Given
  M  Be A Differentiable Mapping And At
1
2
Isomorphism Then

2
d  ( v )   ( M )
By
Be Such d  : T
p  M1
p
M
1
Is
  T
Line

M
2
Of
 Is An
Is A Local Homeomorphism
Theorem 2.8.3
Let G Be Lie Group Of Matrices And Suppose That Log Defines A Coordinate Chart The Near The
Identity Element Of G , Identify The Tangent Space g  T G  At The Identity Element With A Linear Subspace
Of Matrices , Via The Log And Then A Lie Algebra With  B , B   B B  B B The Space g Is Called The Lie
Algebra Of G .
Proof:
It Suffices To Show That For Every Two Matrices B , B  g The  B , B  Is Also An Element
Of g As  B , B  Is
Clearly
Anti
Commutative
And
The
(Jacobs
Identity)
Holds
For A ( t )  ( B t ) ( B t ) (  B t ) (  B t )
. Define For t    With Sufficiently Small  A Path A (T ) In G Such
1
1
2
1
1
1
2
2
1
1
2
2
1
That A ( O )
(22)
2
 I
exp
2
exp
1
exp
2
exp
Using For Each Factor The Local Formula
( Bt ) exp  I  Bt  1 / 2 B t
DOI: 10.9790/5728-1221121136
2
2
 O ( t ) A ( t )  I  B 1 , B 2  t
2
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2
 O (t ) , t  0
131 | Page
Applications Geometry Riemannian Manifolds
Hence B ( t )
 log A ( t )   B 1 , B 2  t
2
 O (t )
2
Expr B ( t ) 
A (t )
Hold
For
Any
Sufficiently
That
Lie
Bracket  B , B   g On Algebra Is An Infinitesimal Version Of The Commutation  g , g   g , g  In The
Corresponding (Lie Group).
Theorem 2.8.4
The Tangent Bundle TM  Has A Canonical Differentiable Structure Making It Into A Smooth 2NDimensional Manifold, Where N=Dim. The Charts Identify Any U  U ( T M )  ( TM ) For An Coordinate
1
1
2
p
1
1
1
1
2
p
Neighborhood U  M , With U  R  That Is Hausdorff And Second Countable Is Called (The Manifold Of
Tangent Vectors)
Definition 2.8.5
A Smooth Vectors Fields On Manifolds M Is Map X : M  TM Such That :(I) X ( P )  T M For
n
p
Every G (Ii) In Every Chart X Is Expressed As a
Coordinates x .
i
( /  xi )
With Coefficients a
i
(x)
Smooth Functions Of The Local
i
III.
Differentiable Manifolds Chart
In This Section, The Basically An M-Dimensional Topological Manifold Is A Topological Space M
Which Is Locally Homeomorphism To R , Definition Is A Topological Space M Is Called An M-Dimensional
(Topological Manifold) If The Following Conditions Hold: (I) M Is A Hausdorff Space.(Ii) For Any
p  M There Exists A Neighborhood U Of P Which Is Homeomorphism To An Open Subset V  R .
(Iii) M Has A Countable Basis Of Open Sets Coordinate Charts ( U ,  ) Axiom (Ii) Is Equivalent To Saying
m
m
p M
Has A Open Neighborhood U  P Homeomorphism To Open Disc D In R , Axiom (Iii) Says
M
Can Covered By Countable Many Of Such Neighborhoods , The Coordinate Chart
( U ,  ) Where U Are Coordinate Neighborhoods Or Charts And  Are Coordinate . A Homeomorphisms ,
Transitions Between Different Choices Of Coordinates Are Called Transitions Maps      , Which Are
That
That
m
m
i j
Again Homeomorphisms By Definition , We Usually Write p  
For U , And p   ( x ) ,  : V  U   M As Coordinates For
1
1
1
( x ) ,  : U  V
U
j

i
As Coordinates
, The Coordinate Charts ( U ,  ) Are
R
n

Coordinate Neighborhoods, Or Charts , And
Are Coordinate Homeomorphisms , Transitions Between
Different Choices Of Coordinates Are Called Transitions Maps      Which Are Again
i j
j
i
Homeomorphisms By Definition , We Usually x   ( p ) ,  : U  V  R As A Parameterization U A
Collection A  ( i , U i )i I Of Coordinate Chart With M   U Is Called Atlas For M .The Transition
Maps  A Topological Space M Is Called ( Hausdorff ) If For Any Pair p , q  M , There Exist Open
n
i
i
i j
Neighborhoods p  U And q  U  Such That U  U    For A Topological Space M With Topology
  U Can Be Written As Union Of Sets In  , A Basis Is Called A Countable Basis  Is A Countable Set .
Definition 3.1.1
A Topological Space M Is Called An M-Dimensional Topological Manifold With Boundary
 M  M If The Following Conditions.
(I) M Is Hausdorff Space.(Ii) For Any Point p  M There Exists A Neighborhood U Of p Which Is
Homeomorphism To An Open Subset V  H .(Iii) M Has A Countable Basis Of Open Sets, Can Be
Rephrased As Follows Any Point p  U Is Contained In Neighborhood U To D  H The Set M Is A
Locally Homeomorphism To R Or H The Boundary  M  M Is Subset Of M Which Consists Of
Points p .
Definition 3.1.2
A Function f : X  Y Between Two Topological Spaces Is Said To Be Continuous If For Every
Open Set U Of Y The Pre-Image f (U ) Is Open In X .
Definition 3.1.3
Let X And Y Be Topological Spaces We Say That X And Y Are Homeomorphism If There Exist
Continuous Function Such That f  g  id And g  f  id We Write X  Y And Say That f And
m
m
m
m
m
1
X
y
Are Homeomorphisms Between
And
, By The Definition A Function f : X  Y Is A
Homeomorphisms If And Only If .(I) f Is A Objective .(Ii) f Is Continuous (Iii) f Is Also Continuous.
3.2 Differentiable Manifolds
A Differentiable Manifolds Is Necessary For Extending The Methods Of Differential Calculus To
Spaces More General R A Subset S  R Is Regular Surface If For Every Point p  S The A Neighborhood
g
X
Y
1
n
DOI: 10.9790/5728-1221121136
3
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Applications Geometry Riemannian Manifolds
Of P Is R And Mapping x : u  R  V  S Open Set U  R Such That. (I) x Is Differentiable
Homomorphism. (Ii) The Differentiable ( dx ) :  R  R  , The Mapping x Is Called A Aparametnzation Of
2
3
V
2
2
3
q
S At P The Important Consequence Of Differentiable Of Regular Surface Is The Fact That The Transition
Also Example Below If x  : U   S And x  : U   S Are x  (U  )  x  (U  )  w   , The
1
1
Maps
1
 x : x
x
1
(w)  R
2
And
1
1
x  x   x  ( w )  R
.
Are Differentiable Structure On A Set M Induces A Natural Topology On M It Suffices To A  M To Be An
Open Set In M If And Only If x  ( A  x  (U  )) Is An Open Set In R For All  It Is Easy To Verify That
M And The Empty Set Are Open Sets That A Union Of Open Sets Is Again Set And That The Finite
Intersection Of Open Sets Remains An Open Set. Manifold Is Necessary For The Methods Of Differential
Calculus To Spaces More General Than De R , A Differential Structure On A Manifolds M Induces A
Differential Structure On Every Open Subset Of M , In Particular Writing The Entries Of An  n  k  Matrix In
1
n
n
Succession Identifies The Set Of All Matrices With
K-Frame That Is Set Of
k
General Linear Group
R
n ,k
Linearly Independent Vectors In
GL ( n )
By The Foregoing
nk
, An
n
R
,
Matrix Of Rank
V n ,k K  n
k
Can Be Viewed As A
Is Called The Steels Manifold ,The
n
Is Differential Structure On The Group
V n ,k
Of
Orthogonal Matrices, We Define The Smooth Maps Function f : M  N Where M , N Are Differential
Manifolds We Will Say That f Is Smooth If There Are Atlases  U  , h   On M ,  V , g  On N , Such
B
B
That The Maps  g f h   Are Smooth Wherever They Are Defined f Is A Homeomorphism If Is Smooth
And A Smooth Inverse. A Differentiable Structures Is Topological Is A Manifold It An Open Covering
U  Where Each Set U  Is Homeomorphism, Via Some Homeomorphism h  To An Open Subset Of
Euclidean Space R , Let M Be A Topological Space , A Chart In M Consists Of An Open Subset
U  M And A Homeomorphism h Of U Onto An Open Subset Of R , A C Atlas On M Is A Collection
 U  , h   Of Charts Such That The U  Cover M And h , h   The Differentiable .
Definition 3.2.1
Let M Be A Metric Space We Now Define What Is Meant By The Statement That M Is An NDimensional C Manifold. (I) A Chart On M Is A Pair ( U ,  ) With U An Open Subset Of M And  A
Homeomorphism A (1-1) Onto, Continuous Function With Continuous Inverse From U To An Open Subset
Of R , Think Of  As Assigning Coordinates To Each Point Of U . (Ii) Two Charts ( U ,  ) And
( V , ) Are Said To Be Compatible If The Transition Functions.
1
B
n
r
m
1
B

n

   


   
(23)

1
:  (U  V )  R

1
:  (U  V )  R
n
  (U  V )  R
n
  (U  V )  R
n
n





Are C That Is All Partial Derivatives Of All Orders Of     And     Exist And Are
Continuous.(Iii) An Atlas For M Is A Family A   (U ,  ) : i  I  Of Charts On M Such That  U  Is
1

i
1
i I
i
i
An Open Cover Of
And Such That Every Pair Of Charts In
Are Compatible. The Index Set I Is
Completely Arbitrary. It Could Consist Of Just A Single Index. It Could Consist Of Uncountable Many Indices.
An Atlas A Is Called Maximal If Every Chart ( U ,  ) On M That Is Compatible With Every Chat Of A .
Example 3.2.2 : Surfaces An N-Dimensional
Any Smooth N-Dimensional R Is An N-Dimensional Manifold. Roughly Speaking A Subset Of
A An
N-Dimensional Surface If , Locally m Of The m  n Coordinates Of Points On The
R
Surface Are Determined By The Other n Coordinates In A C Way , For Example , The Unit Circle S Is A
A
M
n 1
nm

One Dimensional Surface In
R
Near
S
(-1.0) ,
( x, y )
Is On
2
1
. Near (0.1) A Point
nm
Dimensional Surface In R
If
Are
z  ( z ,..., z
)  M There
1  J 1  j 2  ...  j n  m C
Point



Function
x  ( x 1 ,...., x n  m )  U
x i1  f i1 x
C
M
A
nm
1
,x
j1
Function
,...., x
j2
f 1 ,..., f m
jn
z
, x
fk (x
j1
( x, y )  R
y  
If And Only If
1 x
Is A Subset Of
Neighborhood
,..., x
jn
1
,k
)

2
2
Is On S If And Only If
1
y 
. The Precise Definition Is That
R
U
nm
z
 1,...,
Of
1 x
M
2
And
Is An N-
With The Property That For Each
n
, And
Integers.
z In R
nm
n  m  /  j1 ,..., j n 
Such
That
The
. That Is We May Express The Part Of
That Is Near z As
 f  x , x ,...., x 
, x  f  x , x ,...., x  . Where There For Some
M
i2
i2
j1
. We Many Use
DOI: 10.9790/5728-1221121136
x
j2
,x
j1
jn
,...., x
j2
im
jn
im
j1
j2
As Coordinates For
www.iosrjournals.org
jn
R
2
In
M U
z
.Of Course An
133 | Page
Applications Geometry Riemannian Manifolds
Atlas Is With
z  M
 z (x)  (x
j1
,..., x
jn
)
Equivalently,
, There Are A Neighborhood
Vector  
gz
( z ) ,1  k  m
U
z
Of
z
M
In
Is An
R
nm
N-Dimensional Surface In
, And

m C
Functions
 Linearly Independent Such That The Point
xU
z
gk :U
Is In
R
nm
If For Each
With The
 R
z
If And Only If
M
Given By The g To The Explicit
g ( x )  0 For All 1  k  m .To Get From The Implicit Equations For
Equations For M Given By The f One Need Only Invoke ( Possible After Renumbering Of x ) . A
Topological Space M Is Called An M-Dimensional Topological Manifold With Boundary  M  M If The
Following Conditions.(I) M Is Hausdorff Space .(Ii) For Any Point p  M There Exists A Neighborhood U Of
p Which Is Homeomorphism To An Open Subset V  H
(Iii) M Has A Countable Basis Of Open Sets, Can
Be Rephrased As Follows Any Point p  U Is Contained In Neighborhood U To D  H The Set M Is A
Locally Homeomorphism To R Or H The Boundary  M  M Is Subset Of M Which Consists Of Points
p .
Definition 3.2.3
Let X Be A Set A Topology U For X Is Collection Of X Satisfying :(I)  And X Are In U .(Ii)
The Intersection Of Two Members Of U Is In U .(Iii) The Union Of Any Number Of Members U Is In U .
The Set X With U Is Called A Topological Space The Members U  u Are Called The Open Sets. Let
X Be A Topological Space A Subset N  X With x  N Is Called A Neighborhood Of x If There Is An
Open Set U With x  U  N , For Example If X A Metric Space Then The Closed Ball D  ( x ) And The
Open Ball D  ( x ) Are Neighborhoods Of x A Subset C Is Said To Closed If X \ C Is Open
Definition 3.2.4
A Function f : X  Y Between Two Topological Spaces Is Said To Be Continuous If For Every
M
k
k
k
m
m
m
m
m
1
Open Set U Of Y The Pre-Image f (U ) Is Open In X .
Definition 3.2.5
Let X And Y Be Topological Spaces We Say That X And Y Are Homeomorphisms If There Exist
Continuous Function f :  X  Y  , g : Y  X  Such That  f  g   id And  g  f   id We Write
y
X
X  Y
And Say That f And g Are Homeomorphisms Between X And Y , By The Definition A
Function f :  X  Y  Is A Homeomorphisms If And Only If
(I) f Is A Objective (Ii) f Is Continuous
1
(Iii) f Is Also Continuous.
3 .3 Differentiable Manifolds
A Differentiable Manifolds Is Necessary For Extending The Methods Of Differential Calculus To
Spaces More General R A Subset S  R Is Regular Surface If For Every Point p  S The A Neighborhood
V Of P Is R And Mapping x : u  R  V  S Open Set U  R Such That: (I) x Is Differentiable
Homomorphism (Ii) The Differentiable ( dx ) : R  R , The Mapping x Is Called Aparametnzation Of
3
n
3
2
2
2
3
q
At P The Important Consequence Of Differentiable Of Regular Surface Is The Fact That The Transition
Also Example Below If x  : U   S And x  : U   S Are x  (U  )  x  (U  )  w   The Mappings
S
1
1
x
1

 x
: x
1
(w)  R
2
, x
1
 x

1
x (w)  R
A Differentiable Manifold Is Locally Homeomorphism To R The Fundamental Theorem On Existence,
Uniqueness And Dependence On Initial Conditions Of Ordinary Differential Equations Which Is A Local
Theorem Extends Naturally To Differentiable Manifolds. For Familiar With Differential Equations Can Assume
The Statement Below Which Is All That We Need For Example X Be A Differentiable On A Differentiable
M And
U
 M An
Manifold
There Exist A Neighborhood
p  M Then
p  M And
n
p
Inter (  
, ) ,  0 ,
t   (t , q )

1
( t )  X ( ( t ))
And
 ( 0, q )  q
And
And 
(0)  q
A Differentiable Mapping  : (   ,  )  U  M Such
 : (   ,  )  M Which
Acurve
Satisfies
The
Is Called A Trajectory Of The Field
X
That Curve
Conditions
That Passes Through q For t
 0
.A
Differentiable Manifold Of Dimension N Is A Set M And A Family Of Injective Mapping x   R  M Of
Open Sets u   R Into M Such That: (I) u  x  ( u  )  M (Ii) For Any  ,  With x  ( u  )  x  ( u  ) (Iii)
n
n
The Family
Is Maximal Relative To Conditions (I),(Ii) The Pair ( u  , x  ) Or The Mapping x  With
p  x  ( u  ) Is Called A Parameterization , Or System Of Coordinates Of M , u  x  ( u  )  M The Coordinate
Charts ( U ,  ) Where U Are Coordinate Neighborhoods Or Charts , And  Are Coordinate Homeomorphisms
(u  , x )
Transitions Are Between Different Choices Of Coordinates Are Called Transitions Maps 
DOI: 10.9790/5728-1221121136
www.iosrjournals.org
i, j

: 
1
j
i
.
134 | Page
Applications Geometry Riemannian Manifolds
x   ( p ) ,  :U  V  R
Which Are Anise Homeomorphisms By Definition, We Usually Write
1
Collection
n
1
And p   ( x ) ,  : V  U  M For Coordinate Charts With Is M   U Called An Atlas For M Of
Topological Manifolds. A Topological Manifold M For Which The Transition Maps   (   ) For All
U
i
i, j
Pairs 
i
,
j
R

Are C Maps Whose Inverses Are Also
   ( 
(24)
i, j
Since 
i
In The Atlas Are Homeomorphisms Is Called A Differentiable , Or Smooth Manifold , The
Transition Maps Are Mapping Between Open Subset Of
m
j
j
C

m
R
, Homeomorphisms Between Open Subsets Of
Maps , For Two Charts U And
U
i
i
1

):  i (U
i
U


)   j (U
j
 U
i
The Transitions Mapping
j
j
)

 And      Are Homeomorphisms It Easily Follows That Which Show That Our Notion Of
Rank Is Well Defined  J f    J      J f       , If A Map Has Constant Rank For All
1
 
1
1
x
1
1
j
y
p N
We Simply Write rk ( f ) , These Are Called Constant Rank Mapping. The Product Two Manifolds
Two C -Manifolds
Of
Dimension n And n Respectively
The
Topological
M And M Be
Space M  M Are Arbitral Unions Of Sets Of The Form U  V Where U Is Open In M And V Is Open
k
1
2
1
2
1
Structure C
In M , Can Be Given The
Any
Charts M On
2
V
1
On M
1
2
1
 M
Where 
2
All  p , q   U
i
V

i
j
:U
i
j
,
V
k
Manifolds Of Dimension n
,
Declare
M We
j
( n1  n 2 )
By Defining Charts As Follows For
That U  V ,     Is
Chart
i
2
 R
j
, n2
1
Is Defined So That 
i

j
k
Compatible With The Atlas A If Every Map 
i
1

 And 
1
i
i
j
 p , q    i ( p )
. A Given A C N-Atlas, A On M For Any Other Chart U
j
j
,
,
j
(q )
 We Say That U
 Is C Whenever U
U
k
 0
i
 For
,
 Is
The Two
~
A
Atlases
And Is Compatible If Every Chart Of One Is Compatible With Other Atlas A Sub Manifolds Of
Others Of R For Instance S Is Sub Manifolds Of R It Can Be Obtained As The Image Of Map Into R Or
As The Level Set Of Function With Domain R We Shall Examine Both Methods Below First To Develop The
Basic Concepts Of The Theory Of Riemannian Sub Manifolds And Then To Use These Concepts To Derive A
Equantitive Interpretation Of Curvature Tensor , Some Basic Definitions And Terminology Concerning Sub
Manifolds, We Define A Tensor Field Called The Second Fundamental Form Which Measures The Way A Sub
Manifold Curves With The Ambient Manifold , For Example X Be A Sub Manifold Of Y Of  : E  X And
g : E  Y Be Two Vector Brindled And Assume That E Is Compressible , Let f : E  Y And g : E  Y Be
Two Tubular Neighborhoods Of X In Y Then There Exists .
Theorem 3.3.1
Let m , n  N And Let U  R
Be An Open Set, Let g : U  R Be C With g ( x , y )  0 For Some
A
n
2
3
3
3
1
1
nm

m
0
x0  R
n
, y0  R
V  R
Sets
(x , y )  V
xW
The
m
With ( x
nm
And W
With g ( x ,
N-Sphere
g ( x 1 , .... , x n  1 ) 
x
 R
n
0
y ) 
S
n
Is
The

As R
(
Because
. S O ( 3 )   3  3 real
Example 3.3.2
0
Cylindrical
1 i , j  m
 0
x 1  ....  x n
2
33
2
Matrix
R , R R  1 , det R  1
t
Coordinates
0
n
2
In R In
( x 0 , y 0 )]
0
If You Think Of The Set Of All 3  3 Real Matrices
Has
9
Matrix
Elements
)
Then

The Torus T Is The Two Dimensional Surface
3
j
N-Dimensional Surface R Given Implicitly By Equation
In A Neighborhood Of , For Example The Northern Hemisphere S Is
A
matrices
y
0
Then There Exist Open
n 1
x n 1 
Given Explicitly By The Equation
9
i
0
2
1
det [
g
With ( x , y )  V Such That , For Each x  W There Is A Unique
0 If The Y Above Is Denoted f  x   y And g  x , f  x    0 For All
 .....  x n  1
2
. Assume That
, y0 )  U
0
T
2

( x , y , z )  R
x  r cos  , y  r sin  , z  0
3
The
,(
x
2
 y
Equation
2
 1)
Of
2
 z
2
 1/ 4
The

Torus
Is ( r  1)  z  1 / 4 Fix Any  , say  . Recall That The Set Of All Points In R That Have    Is An
Open Book, It Is A Hall-Plane That Starts At The z Axis. The Intersection Of The Tours With That Half Plane
Is Circle Of Radius 1/2 Centered On r  1 , z  0 As  Runs Form 0 to 2  , The Point
r  1  1 / 2 cos  And
0 to 2  The
   Runs Over That Circle. If We Now Run
 From
Point ( x , y , z )  ( (1  1 / 2 cos  ) cos  , ( 1  1 / 2 sin  ) Runs Over The Whole Torus. So We May Build
2
n
2
0
0
0
Coordinate Patches For
T
2
Using
DOI: 10.9790/5728-1221121136

And

With Ranges ( 0 , 2  ) Or
(   , )
www.iosrjournals.org
As Coordinates)
135 | Page
Applications Geometry Riemannian Manifolds
Definition 3.3.3
(I) A Function
f
From A Manifold M To Manifold
The Notation) Is Said To Be
C

At
m M
N
(It Is Traditional To Omit The Atlas From
If There Exists A Chart 
U ,   For M
And Chart 
V ,
 For
And    f    Is C At  ( m ) . (Ii) Tow Manifold M And N Are
Diffeomorphic If There Exists A Function f : M  N That Is (1-1) And Onto With N And f On
C Everywhere. Then You Should Think Of M And N As The Same Manifold With m And f ( m ) Being
Two Names For Same Point, For Each m  M .
Such That
N
1
m  U , f (m )  v

1

IV.
Conclusion
The Basic Notions On Applications Geometry Riemannian Knowledge Of Calculus Manifolds,
Including The Geometric Formulation Of The Notion Of The Differential And The Inverse Function Theorem.
The Differential Geometry Of Surfaces With The Basic Definition Of Differentiable Manifolds, Starting With
Properties Of Covering Spaces And Of The Fundamental Group And Its Relation To Covering Spaces.
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www.iosrjournals.org
136 | Page