Differential Geometry and Relativity Theories

Differential Geometry and Relativity Theories vol. 1 Tangent vectors, derivatives, paths, 1-forms

David Carfì

Editions Il Gabbiano 2016

Differential Geometry and Relativity Theories vol. 1 Tangent vectors, tangent maps, paths, 1-forms

c David Carfì 2016 ○ [email protected]

Editions il Gabbiano 2016 di Maria Froncillo Nicosia ISBN 978-88-96293-22-5 printed by “Centro Tesi” di Aldo Lo Faro via Giacomo Venezian 98122 Messina

to Maria Jose Di Marco. Poet, writer, music critic, journalist, lawyer. 25 December 1949 - 2 February 2001.

Contents 0 Introduction

5

1 Charts, atlases, manifolds

9

1.1 1.2 1.3 1.4 1.5 1.6 1.7 1.8 1.9

I

Charts and atlases . . . . . . . . . Transition maps . . . . . . . . . . . Manifolds . . . . . . . . . . . . . . Compatibility of charts and atlases Dimension of a manifold . . . . . . Coordinate systems . . . . . . . . . Smooth manifolds . . . . . . . . . . Induced differentiable structures . . Smooth functions . . . . . . . . . .

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Tangent vectors

17

2 Tangent vectors 2.1 2.2 2.3 2.4

9 10 10 11 12 12 13 13 13

Definition of tangent vectors . . . Some remarks . . . . . . . . . . . Basic properties . . . . . . . . . . Example: tangent vectors upon R 1

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19 21 22 24

CONTENTS

3 Tangent vectors and charts

27

4 Tangent frames

33

3.1 3.2 3.3 4.1 4.2 4.3 4.4

II

Change of coordinate system . . . . . . . . . . . . . . . Action of tangent vectors on charts . . . . . . . . . . . Structure of tangent spaces . . . . . . . . . . . . . . . . Geometric interpretation . . . . . . . . Tangent vectors determined by charts . Tangent frames induced by local charts Change of local frames . . . . . . . . .

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Contravariant and covariant vectors

5 Contravariant vectors 5.1 5.2 5.3 5.4 5.5 5.6

Introduction . . . . . . . . . . . . Definition . . . . . . . . . . . . . Contravariant vector components Contravariant vector space . . . . Contravariant 2-tensors . . . . . . Tensor product . . . . . . . . . .

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27 28 30

33 35 37 39

43

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45 46 47 48 50 51

6 Covariant vectors

53

III

61

6.1 6.2 6.3

Covariant vectors . . . . . . . . . . . . . . . . . . . . . The gradient of a real function . . . . . . . . . . . . . . Invariant scalars . . . . . . . . . . . . . . . . . . . . . .

Derivation

7 Derivatives 7.1 7.2

Introduction . . . . . . . . . . . . . . . . . . . . . . . . Definition of derivatives . . . . . . . . . . . . . . . . . 2

53 55 56

63

63 65

CONTENTS

7.3 7.4

Local Jacobian matrices . . . . . . . . . . . . . . . . . Basic properties of the tangent map . . . . . . . . . . .

66 68

8 Chain rules and diffeomorphisms

69

IV

75

8.1 8.2 8.3

Chain rule . . . . . . . . . . . . . . . . . . . . . . . . . Diffeomorphisms and derivatives . . . . . . . . . . . . . Tangent bases under derivatives . . . . . . . . . . . . .

Paths on manifolds

9 Paths on manifolds 9.1 9.2 9.3 9.4 9.5

Paths on a Manifold . . . . . . . . . Derivatives by re-parametrizations . . Tangent vector vs calculus derivative An elementary example . . . . . . . . Vector derivative in local coordinates

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69 70 72

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77 79 80 82 83

10 Paths and tangent vectors

85

V

91

10.1 Path initial tangent vector . . . . . . . . . . . . . . . . 10.2 Tangent vectors as vector derivatives . . . . . . . . . . 10.3 Derivatives and paths . . . . . . . . . . . . . . . . . . . Covectors

11 Covectors and differentials 11.1 11.2 11.3 11.4 11.5 11.6

Cotangent vectors . . . . . . . . Differential 1-forms . . . . . . . Differential of a function . . . . Exact 1-forms . . . . . . . . . . Derivative versus differentials . Critical points of real functions 3

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85 87 89

93

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CONTENTS

11.7 Differential of a vector function . . . . . . . . . . . . . 101 11.8 Derivative versus differential bis . . . . . . . . . . . . . 102

12 Covector frames

12.1 Change of covector frames . . . . 12.2 Generalized contravariant vectors 12.2.1 Einstein’s cotangent bases 12.3 Generalized covariant vectors . .

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105 105 107 108 109

Chapter 0 Introduction In this book, we focus on some aspects of smooth manifolds, which appear of fundamental importance for the developments of differential geometry and its applications to Theoretical Physics, Special and General Relativity, Economics and Finance. In particular we touch basic topics, for instance: 1. definition of tangent vectors; 2. change of coordinate system in the definition of tangent vectors; 3. action of tangent vectors on coordinate systems; 4. structure of tangent spaces; 5. geometric interpretation of tangent vectors; 6. canonical tangent vectors determined by local charts; 7. tangent frames determined by local charts; 8. change of local frames; 5

9. tangent vectors and contravariant vectors; 10. covariant vectors; 11. the gradient of a real function; 12. invariant scalars; 13. tangent applications; 14. local Jacobian matrices; 15. basic properties of the tangent map; 16. chain rule; 17. diffeomorphisms and derivatives; 18. transformation of tangent bases under derivatives; 19. paths on a manifold; 20. vector derivative of a path with respect to a re-parametrization; 21. tangent derivative versus calculus derivative; 22. vector derivative of a path in local coordinates; 23. existence of a path with a given initial tangent vector; 24. tangent vectors as vector derivatives of paths; 25. derivatives and paths; 26. cotangent vectors; 27. differential 1-forms; 6

CHAPTER 0.

28. differential of a function; 29. derivative versus differentials; 30. critical points of real functions; 31. differential of a vector function; 32. change of covector frames; 33. generalized contravariant vectors; 34. generalized covariant vectors.

7

INTRODUCTION

8

Chapter 1 Charts, atlases, manifolds 1.1 Charts and atlases We begin with the concept of local chart.

Definition (of local chart). 𝑆. subset 𝑈 space

A local chart

𝑥

on

𝑆

of the topological space

Euclidean space

Consider a Hausdorff topological

is a topological isomorphism of an open

𝑆

onto an open subset

𝑥(𝑈 )

of some

𝐸𝑥 .

Now, we can define the fundamental object of atlas.

Definition (of atlas). An atlas of class cal space

𝑆

𝐶 𝑘 (𝑘

Consider a Hausdorff topological space

𝑆.

positive integer or infinity) on the topologi-

is a collection

𝒜

of charts on

𝑆

satisfying the following

conditions:

∙

the collection of all chart domains of the atlas logical space

𝑆; 9

𝒜 covers the topo-

1.2.

TRANSITION MAPS

∙

for any charts

𝑥, 𝑦

of the atlas

𝒜,

with domain

𝑈, 𝑉

respectively,

the map

𝑦/𝑥 : 𝑥(𝑈 ∩ 𝑉 ) → 𝑦(𝑈 ∩ 𝑉 ) defined by

𝑥0 ↦→ (𝑦 ∘ 𝑥− )(𝑥0 ) is a

𝐶 𝑘 -isomorphism (𝐶 𝑘 -diffeomorphism,

if

𝑘

is different from

0 or infinity).

1.2 Transition maps The above map 𝑦/𝑥 is denoted also by 𝑦𝑥− and called the transition map from the chart 𝑥 to the chart 𝑦 . It can also be defined by

𝑥(𝑝) ↦→ 𝑦(𝑝), for every 𝑝 in the intersection of 𝑈 and 𝑉 . Observe that, for any choice of charts 𝑥, 𝑦 of the atlas 𝒜 - with domain 𝑈, 𝑉 , respectively - the images

𝑥(𝑈 ∩ 𝑉 ), 𝑦(𝑈 ∩ 𝑉 ), reveal open in the Euclidean spaces 𝐸𝑥 and 𝐸𝑦 , because the charts 𝑥, 𝑦 are homeomorphisms.

1.3 Manifolds In the conditions of the preceding definition of atlas the pair (𝑆, 𝒜) is called a manifold of class 𝐶 𝑘 , if no other atlas contains properly 𝒜. 10

CHAPTER 1.

CHARTS, ATLASES, MANIFOLDS

Let 𝑀 be a manifold with atlas 𝒜. Each pair (𝑈, 𝑥), with 𝑥 in 𝒜 and 𝑈 domain of 𝑥 will be called also a chart of the atlas. If a point 𝑝 of 𝑀 lies in 𝑈 , then we say that 𝑥 (and also (𝑈, 𝑥)) is a chart of the manifold 𝑀 at the point 𝑝. In the definition of atlas, we did not require that the Euclidean spaces be the same for all charts 𝑥. If they are all equal to the same space 𝐸 , then we say that the atlas is an 𝐸 -atlas. If two charts (𝑈, 𝑥) and (𝑉, 𝑦) are such that 𝑈 and 𝑉 show a nonempty intersection, and if the class 𝑘 is strictly greater than 1, then - taking the derivative of the transition map 𝑦/𝑥 - we see that the Euclidean spaces 𝐸𝑥 and 𝐸𝑦 are linearly isomorphic, then equal. Furthermore, the set of points 𝑝 of the topological space 𝑆 , for which there exists a chart 𝑥 at the point 𝑝 such that 𝐸𝑥 is linearly isomorphic to a given space 𝐸 is both open and closed. Consequently, on each connected component of the manifold 𝑀 , we could assume that we have an 𝐸 -atlas for some fixed space 𝐸 .

1.4 Compatibility of charts and atlases Consider an open subset 𝑉 of a topological space 𝑆 and a topological isomorphism 𝑦 : 𝑉 → 𝑦(𝑉 ) onto an open subset of some Euclidean space 𝐸 . We shall affirm that the chart 𝑦 is compatible with the atlas 𝒜 if each transition map 𝑦/𝑥 (defined on the convenient intersection as in the atlas definition) is a 𝐶 𝑘 -isomorphism. 11

1.5.

DIMENSION OF A MANIFOLD

Two atlases are said to be compatible if each chart of one is compatible with the other atlas.

We can verify immediately that the relation of compatibility between atlases is an equivalence relation. An equivalence class of atlases of class 𝑘 structure of 𝐶 - manifold on 𝑆 .

𝐶𝑘

on

𝑆

is said to define a

A 𝑘 -differentiable structure on a topological space defines a maximal atlas on that space.

1.5 Dimension of a manifold If all the Euclidean spaces 𝐸𝑥 in some atlas 𝒜 are linearly isomorphic, then they are all equal, say to the Euclidean space 𝐸 . We then say that the manifold 𝑀 = (𝑆, 𝒜) is an 𝐸 -manifold or that 𝑀 is modeled on 𝐸 , or that 𝑀 is an 𝑚-manifold, if 𝐸 is the 𝑚-dimensional Euclidean space. In other terms, under the above conditions, if 𝐸 reveals equal to R𝑚 , for some fixed 𝑚, then we say that the manifold 𝑀 is 𝑚dimensional.

1.6 Coordinate systems In 𝑚-dimensional manifolds, all charts

𝑥 : 𝑈 → 𝑥(𝑈 ) 12

CHAPTER 1.


are defined by an ordered system of 𝑚 coordinate functions:

(𝑥𝑖 )𝑚 𝑖=1 . If 𝑝 denotes a point of the domain 𝑈 of 𝑥, the coordinate system of 𝑝 in the chart 𝑥 is the family

(𝑥𝑖 (𝑝))𝑚 𝑖=1 . The charts 𝑥 themselves are also called local coordinate systems of the manifold.

1.7 Smooth manifolds If the extended integer 𝑘 (which may also be infinity) is fixed throughout a discussion, we also simply say that 𝑀 is a manifold. A smooth manifold is a manifold with class of differentiability equal to infinity.

1.8 Induced differentiable structures Let 𝑀 be a manifold, and 𝑉 an open subset of 𝑀 (that is an open subset of the underlined topological space). Then, it is possible, in the obvious way, to induce a manifold structure on the open subset 𝑉 , by taking as charts (of the induced structure) the restrictions of the charts of 𝑀 to the subset 𝑉 .

1.9 Smooth functions Let 𝑀 be a smooth manifold of dimension 𝑚. A vector function

𝑓 : 𝑀 → R𝑛 13

1.9.

SMOOTH FUNCTIONS

is said to be smooth at a point 𝑝 in 𝑀 , if there exists a chart (𝑈, 𝑥) about 𝑝 in 𝑀 such that

𝑓/𝑥 =: 𝑓 ∘ 𝑥− , a function defined on the open subset 𝑥(𝑈 ) of R𝑚 , is smooth at 𝑥(𝑝). The function 𝑓 is said to be smooth on 𝑀 if it is smooth at every point 𝑝 of 𝑀 . The definition of the smoothness of a vector function 𝑓 at a point 𝑝 is independent of the chart 𝑥. Let 𝑀 and 𝑁 be manifolds of dimension 𝑚 and 𝑛, respectively. A continuous map 𝑓 :𝑀 →𝑁 is smooth, at a point 𝑝 of 𝑀 if there exist charts (𝑉, 𝑦) about 𝑓 (𝑝) in 𝑁 and (𝑈, 𝑥) about the point 𝑝 of 𝑀 such that the composition 𝑦 𝑓/𝑥

:= 𝑦 ∘ 𝑓 ∘ 𝑥− ,

which is a map from the open subset

𝑥(𝑓 − (𝑉 ) ∩ 𝑈 ) of R𝑚 to the open subset 𝑦(𝑉 ) of R𝑛 , is smooth at the point 𝑥(𝑝). The continuous map

𝑓 :𝑀 →𝑁 is said to be smooth if it is smooth at every point of 𝑀 . 14

CHAPTER 1.


The above definition of smoothness is equivalent to assert that a continuous map 𝑓 :𝑀 →𝑁 is smooth, at a point 𝑝 of 𝑀 if there exists a chart (𝑉, 𝑦) about the image point 𝑓 (𝑝) of the codomain manifold 𝑁 such that the composition 𝑦 𝑓 := 𝑦 ∘ 𝑓, a map from the open subset 𝑓 − (𝑉 ) of 𝑀 to R𝑛 , is smooth at the point 𝑝.

15

1.9.

SMOOTH FUNCTIONS

16

Part I Tangent vectors

17

Chapter 2 Tangent vectors 2.1 Definition of tangent vectors Here, we introduce the fundamental concept of a tangent vector on differentiable (smooth) manifolds.

Definition (of tangent vector). fold

𝑀,

with dimension

𝑚,

Consider a smooth real mani-

and the collection

𝐶𝑝∞ (𝑀, R), of all smooth real functions locally defined upon open neighborhoods of

𝑀 . We define a tangent vector on the manifold 𝑀 at a point 𝑝 of 𝑀 as a real functional a point

𝑝

in

𝑣 : 𝐶𝑝∞ (𝑀, R) → R such that, for each local coordinate system exists a vector

𝑣𝑥 ,

𝑥

of the manifold, there

belonging to the real Euclidean

the value of the functional

𝑣,

𝑚-space,

such that

at any locally defined smooth function

19

𝑓

2.1.

at

𝑝,

DEFINITION OF TANGENT VECTORS

equals the directional derivative

𝜕𝑣𝑥 (𝑓/𝑥 )(𝑥(𝑝)), with respect to the vector

𝑣𝑥 ,

𝑥-representation

of the

𝑓/𝑥 := 𝑓 ∘ 𝑥− , of the real function

𝑓,

at the point

𝑥(𝑝).1

In other terms, let 𝒜 represent the atlas of 𝑀 and let 𝒜𝑝 represent the collection of charts around 𝑝. Then, the above functional 𝑣 is defined a tangent vector at the point 𝑝, if for every chart 𝑥 in 𝒜𝑝 , there exists a vector 𝑣𝑥 of the Euclidean space R𝑚 such that, for any function 𝑓 of 𝐶𝑝∞ (𝑀 ), we see:

𝑣(𝑓 ) = 𝜕𝑣𝑥 (𝑓 /𝑥)(𝑥(𝑝)); or, equivalently,

𝑣(𝑓 ) = 𝑑𝑥(𝑝) (𝑓/𝑥 )(𝑣𝑥 ) = = (𝑓/𝑥 )′𝑥(𝑝) (𝑣𝑥 ), in terms of differentiation and derivation. 1 We

recall that the representation of a smooth function 𝑓 :𝑉 →R

at a point 𝑝, with respect to a chart 𝑥 : 𝑈 → 𝑥(𝑈 )

at the same point 𝑝 appears well-defined by 𝑓/𝑥 := 𝑓 ∘ 𝑥− : 𝑥(𝑈 ∩ 𝑉 ) → R : 𝑥0 ↦→ 𝑓 (𝑥− (𝑥0 )).

Observe that the intersection 𝑈 ∩ 𝑉 reveals an open neighborhood of 𝑝.

20

CHAPTER 2.

TANGENT VECTORS

2.2 Some remarks We note, preliminary (and we shall underline again this important aspect of tangent vector definition), that we can extend immediately the action of a tangent vector to any vector function

𝑓 : 𝑉 → R𝑛 , locally defined at 𝑝, by putting

𝑣(𝑓 ) = 𝑑𝑥(𝑝) (𝑓/𝑥 )(𝑣𝑥 ), using again the representation

𝑓/𝑥 = 𝑓 ∘ 𝑥− , of 𝑓 by 𝑥. It appears clear that the image 𝑣(𝑓 ) belongs to the Euclidean space R𝑛 , as the linear operator

𝑑𝑥(𝑝) (𝑓/𝑥 ) : R𝑚 → R𝑛 takes values in R𝑛 . This extension does not depend on the particular choice of the chart 𝑥. Indeed, component-wise, we see immediately 𝑗 𝑑𝑥(𝑝) (𝑓/𝑥 )(𝑣𝑥 ) = [𝑑𝑥(𝑝) (𝑓/𝑥 )(𝑣𝑥 )]𝑛𝑗=1 = [𝑣(𝑓 𝑗 )]𝑛𝑗=1 ,

for any chart 𝑥 at the point 𝑝. As a first useful consequence of the above extension, we observe, for any chart (𝑈, 𝑥), that

𝑣(𝑥) = = = =

𝑑𝑥(𝑝) (𝑥/𝑥 )(𝑣𝑥 ) = 𝑑𝑥(𝑝) (I𝑥(𝑈 ) )(𝑣𝑥 ) = IR𝑚 (𝑣𝑥 ) = 𝑣𝑥 . 21

2.3.

BASIC PROPERTIES

Consequently, the vector 𝑣𝑥 , of the above definition, reveals unique and equals the image 𝑣(𝑥), of the chart 𝑥 under the tangent vector 𝑣 . We shall call the vector 𝑣(𝑥) the representation of 𝑣 by 𝑥. Furthermore, we can so define a mapping

𝑣¯ : 𝒜𝑝 → R𝑚 : 𝑥 ↦→ 𝑣(𝑥), sending each chart in the representation of the tangent vector 𝑣 under 𝑥: the so called contravariant vector of 𝑣 .

2.3 Basic properties Theorem. function

𝑓

The value of a tangent vector

𝑣

𝑝, upon a smooth 𝑓 , that is, if two neighborhood of the point 𝑝 at a point

depends only on the germ of the function

smooth functions

𝑓

and

𝑔

coincide on a

then we obtain

𝑣(𝑓 ) = 𝑣(𝑔).

Proof.

Indeed, the representations 𝑓 /𝑥 and 𝑔/𝑥, in any chart 𝑥 about 𝑝, will coincide on a neighborhood of the point 𝑥(𝑝) and, then, their directional derivatives at 𝑥(𝑝) will be equal. Moreover,

Theorem.The value of a tangent vector at 𝑝 on a test function at 𝑝,

which remains constant around the point

𝑝

is 0.

Proof. Indeed, any representation 𝑓 /𝑥 will be constant about 𝑥𝑝 and then the directional derivative 𝑣𝑓 should be 0. 22

CHAPTER 2.

Theorem.

TANGENT VECTORS

Tangent vectors are linear functional and satisfy the

pointed Leibniz rule:

𝑣(𝑓 𝑔) = 𝑣(𝑓 )𝑔(𝑝) + 𝑓 (𝑝)𝑣(𝑔), for every couple

Proof.

𝑓, 𝑔

of smooth functions.

The proof is straightforward and we leave it as an exercise.

Theorem. fold

𝑀,

Any tangent vector

𝑣

at a point

𝑝,

on a smooth mani-

defines a linear functional (compact support distribution)

𝑢 : 𝐶 ∞ (𝑀, R) → R : 𝑓 ↦→ 𝑣(𝑓 ), restriction of the tangent vector

𝑣

to the vector space

𝐶 ∞ (𝑀, R), acting as a directional derivative in any local coordinate chart at

𝑝.

Vice versa, any linear functional

𝑢 : 𝐶 ∞ (𝑀, R) → R, 𝐶 ∞ (𝑀, R), acting as a directional derivachart at 𝑝, defines a tangent vector 𝑣 such

on the smooth function space tive in any local coordinate that

𝑣(𝑓 ) = 𝑢(𝑔), for any smooth locally defined function

𝑔

𝑓

at

𝑝 𝑓

and any smooth glob-

𝑝 (i.e., any smooth globally defined function 𝑔 which equals the function 𝑓 on a convenient neighborhood of 𝑝). ally defined function

with the same germ of

23

at

2.4.

EXAMPLE: TANGENT VECTORS UPON

R

Proof.

Indeed, let 𝑓 be a smooth real function defined on an open neighborhood 𝑉 of a point 𝑝 and let 𝑓˜ one its extension to the whole of 𝑀 (possibly not smooth...). We want to extend the action of 𝑢 to 𝑓 . We can find a smooth function 𝑔 defined upon 𝑀 such that:

∙ it equals zero out of an open neighborhood 𝑈 whose closure 𝑈¯ is contained in 𝑉 ; ∙ it equals 1 on a compact neighborhood 𝐾 of 𝑝 contained in 𝑈 . Then, the point-wise product 𝑓˜𝑔 reveals a smooth function on the entire manifold 𝑀 , with the same germ of 𝑓 . In this condition, we define the value of 𝑢 on 𝑓 as

𝑢(𝑓 ) := 𝑢(𝑓˜𝑔), so we can define a corresponding tangent vector by

𝑣(𝑓 ) := 𝑢(𝑓˜𝑔), for every 𝑓 locally defined at 𝑝. Clearly, the definition is well posed, because it does not depend on the arbitrate choice of the extension 𝑓˜ nor on the test function 𝑔 , but only upon the germ of 𝑓 at 𝑝.

2.4 Example: tangent vectors upon R Consider a real number 𝑝 and the point derivative functional

𝐷𝑝 : 𝐶𝑝∞ (R, R) → R, on the real line, sending each smooth real function 𝑓 - locally defined on the real line, around the point 𝑝 - to its derivative 𝑓 ′ (𝑝) at the point 𝑝, that is 𝐷𝑝 (𝑓 ) = 𝑓 ′ (𝑝), 24

CHAPTER 2.

TANGENT VECTORS

for every differentiable function 𝑓 . For every coordinate system 𝑥 around 𝑝, we easily obtain that

𝐷𝑝 (𝑓 ) = 𝜕𝑥′ (𝑝) (𝑓 /𝑥)(𝑥(𝑝)), for every differentiable function 𝑓 . Indeed,

𝜕𝑥′ (𝑝) (𝑓 /𝑥)(𝑥(𝑝)) = = = = =

𝑥′ (𝑝)𝜕1 (𝑓 /𝑥)(𝑥(𝑝)) = 𝑥′ (𝑝)(𝑓 /𝑥)′ (𝑥(𝑝)) = 𝑥′ (𝑝)(𝑓 ′ ∘ 𝑥− )(𝑥(𝑝))(𝑥− )′ (𝑥(𝑝)) = 𝑥′ (𝑝)𝑓 ′ (𝑥− (𝑥(𝑝))𝑥′ (𝑥− (𝑥(𝑝)))−1 = 𝑓 ′ (𝑝),

as we claimed. So, 𝐷𝑝 acts like a point directional derivative with respect to each coordinate system around 𝑝. If r represents the identity function on the real line, we can denote the point derivative functional 𝐷𝑝 also by 𝑑/𝑑r|𝑝 . Note that any point derivation 𝑎𝐷𝑝 acts like a directional derivative, indeed 𝑎𝐷𝑝 (𝑓 ) = 𝜕𝑎𝑥′ (𝑝) (𝑓/𝑥 )(𝑥(𝑝)), for every smooth function 𝑓 . An alternative, shorter (but more sophisticated) proof follows:

𝜕𝑥′ (𝑝) (𝑓 /𝑥)(𝑥(𝑝)) = = = = =

𝑥′ (𝑝)𝜕1 (𝑓 /𝑥)(𝑥(𝑝)) = 𝑥′ (𝑝)(𝜕𝑓 /𝜕𝑥)𝑝 = 𝑥′ (𝑝)(𝑑𝑓 /𝑑𝑥)(𝑝) = 𝑥′ (𝑝)(𝑓 ′ (𝑝)/𝑥′ (𝑝)) = 𝑓 ′ (𝑝),

where we have used: 25

2.4.

EXAMPLE: TANGENT VECTORS UPON

R

∙ the definition of the derivative 𝑑𝑓 /𝑑𝑔 := 𝑓 ′ /𝑔 ′ , of a function 𝑓 with respect to another function 𝑔 ;

∙ the property that the derivative 𝜕𝑓 /𝜕𝑥|𝑝 := (𝑓/𝑥 )′ (𝑥(𝑝)), of a function 𝑓 with respect to a chart 𝑥 at 𝑝, equals the derivative (𝑑𝑓 /𝑑𝑥)(𝑝) (we leave the proof to readers).

26

Chapter 3 Tangent vectors and charts 3.1 Change of coordinate system We shall see here that, in order to verify the tangency property of a tangent functional, we need not to verify the tangency property itself for any coordinate system but it just suffices to see it for only one of them. Note, at this purpose, the following basic property.

Property.

If a functional

𝑣 : 𝐶𝑝∞ (𝑀, R) → R acts like a directional derivative in a coordinate system

𝑥

at a point

then it acts like a directional derivative in all coordinate systems at In other terms, if

𝑣(𝑓 ) = 𝑑𝑥(𝑝) (𝑓 /𝑥)(𝑣(𝑥)), then

𝑣(𝑓 ) = 𝑑𝑦(𝑝) (𝑓 /𝑦)(𝑣(𝑦)), 27

𝑝, 𝑝.

3.2.

ACTION OF TANGENT VECTORS ON CHARTS

for any other chart

𝑥, 𝑦

at

𝑝,

𝑦.

Moreover, for any pair of coordinate system

we see

𝑣(𝑦) = 𝑑𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥)), for what concerns the representatives of

𝑣

in the two coordinate charts.

Proof.

Indeed, for any pair of coordinate system 𝑥, 𝑦 at 𝑝, we easily obtain

𝑣(𝑓 ) = = = = =

𝑑𝑥(𝑝) (𝑓 /𝑥)(𝑣(𝑥)) = 𝑑𝑥(𝑝) [(𝑓 /𝑦) ∘ (𝑦/𝑥)](𝑣(𝑥)) = [𝑑𝑦(𝑝) (𝑓 /𝑦) ∘ 𝑑𝑥(𝑝) (𝑦/𝑥)](𝑣(𝑥)) = 𝑑𝑦(𝑝) (𝑓 /𝑦)(𝑑𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥))) = 𝑑𝑦(𝑝) (𝑓 /𝑦)(𝑣(𝑦)),

where we put

𝑣(𝑦) = 𝑑𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥)). As we claimed.

3.2 Action of tangent vectors on charts In a certain sense, we have already extended the domain of action of the tangent functional 𝑣 - by adding, to the collection of locally defined smooth real functions at 𝑝, the collection of all local coordinate systems about 𝑝 - when we decided to define the representatives 𝑣𝑥 of 𝑣 at any coordinate system 𝑥. We shall see in a moment that this extension appears totally compatible with the action of 𝑣 on local defined real functions. 28

CHAPTER 3.

TANGENT VECTORS AND CHARTS

Let us note that the 𝑖-th component 𝑥𝑖 of a coordinate system 𝑥 is a locally defined real function on 𝑀 . We immediately obtain

𝑣(𝑥𝑖 ) = 𝜕𝑣(𝑥) (𝑥𝑖 /𝑥)(𝑥(𝑝)) = = 𝑑𝑥(𝑝) (e𝑖|𝑥 )(𝑣(𝑥)) = = e𝑖 (𝑣(𝑥)) = = 𝑣(𝑥)𝑖 , for any index 𝑖 of 𝑥, where

e𝑖 : 𝑤 ↦→ 𝑤𝑖 represents the 𝑖-projection of the Euclidean 𝑚-space and

e𝑖|𝑥 : 𝑥(𝑈 ) → 𝑥𝑖 (𝑈 ) represents its restriction from 𝑥(𝑈 ) to 𝑥𝑖 (𝑈 ). Therefore, we obtain 𝑚 𝑚 𝑣(𝑥) = (𝑣(𝑥)𝑖 )𝑖=1 = (𝑣(𝑥𝑖 ))𝑖=1 .

Concluding, we can state the following proposition.

Proposition. system

𝑣

𝑥

on the

𝑣 on a coordinate 𝑚-tuple of the single actions of the tangent vector 𝑖 components 𝑥 of the coordinate system itself. The action of the tangent vector

equals the

From the above property we deduce again the uniqueness of vector 𝑣(𝑥), belonging to the real Euclidean 𝑚-space, such that the value of the functional 𝑣 , at any locally defined smooth real function 𝑓 , equals the directional derivative, with respect to 𝑣(𝑥), of the 𝑥-representation

𝑓/𝑥 := 𝑓 ∘ 𝑥− 29

3.3.

STRUCTURE OF TANGENT SPACES

of the real function 𝑓 , at the point 𝑥(𝑝), that is:

𝑣(𝑓 ) = 𝜕𝑣(𝑥) (𝑓 /𝑥)(𝑥(𝑝)).

Definition (representative of a tangent vector). ditions of the preceding definition, we call

In the con-

representative of 𝑣 , in

the system of coordinates 𝑥, the 𝑚-tuple 𝑣(𝑥).

We can extend the action of a tangent vector even more: to vector function locally defined on 𝑀 .

Definition (action of a tangent vector 𝑣 on a vector function). The action of a tangent vector 𝑣 at 𝑝, of a manifold 𝑀 , on a vector function,

𝑓 : 𝑉 → 𝑊, locally defined at

𝑝,

where

𝑉

and

manifold and of the Euclidean actions of the tangent vector function

𝑊

represent two open subsets of the

𝑛-space, equals the 𝑛-tuple of the single 𝑣 on the components 𝑓 𝑗 of the vector

𝑓 𝑛 𝑛 𝑣(𝑓 ) = (𝑣(𝑓 )𝑗 )𝑗=1 = (𝑣(𝑓 𝑗 ))𝑗=1 .

.

3.3 Structure of tangent spaces Consider a manifold 𝑀 and a tangent vector 𝑣 of the manifold to a point 𝑝. The point 𝑝 is called the foot-point of the tangent vector 𝑣 and the collection 𝑀𝑝 of all tangent vectors at 𝑝 is called the tangent space of the manifold 𝑀 at the point 𝑝. That collection is, indeed, a real vector space under the operations of point-wise addition and scalar multiplication, defined by

(𝑣 + 𝑤)(𝑓 ) = 𝑣(𝑓 ) + 𝑤(𝑓 ) 30

CHAPTER 3.

TANGENT VECTORS AND CHARTS

and

(𝑎𝑣)(𝑓 ) = 𝑎𝑣(𝑓 ), for every couple of tangent vectors 𝑣, 𝑤 and every smooth locally defined function 𝑓 . Now, we state and prove a theorem regarding, more specifically, the algebraic structure of tangent spaces of smooth manifolds. At this aim, we introduce here the fundamental concept of differential 𝑑𝑝 𝑥 of a chart 𝑥 at a point 𝑝 of its domain, defined as the below linear operator associating with any tangent vector at the point 𝑝 its representation in the fixed chart 𝑥,

𝑑𝑝 𝑥 : 𝑀𝑝 → R𝑚 : 𝑣 ↦→ 𝑣𝑥 . It appears as a simple and straightforward generalization of the differential 𝑑𝑝 𝑓 of a smooth real function 𝑓 at a point 𝑝 of its domain, defined as the below linear functional associating with any tangent vector 𝑣 at the point 𝑝 its value 𝑣(𝑓 ) upon the fixed smooth function 𝑓, 𝑑𝑝 𝑓 : 𝑀𝑝 → R : 𝑣 ↦→ 𝑣(𝑓 ).

Theorem (structure of tangent spaces). chart

𝑥

of a smooth manifold

𝑀,

Fixed a coordinate

the application

𝑑𝑝 𝑥 : 𝑀𝑝 → R𝑚 : 𝑣 ↦→ 𝑣𝑥 , reveals a linear isomorphism, so that, the algebraic dimension of tangent space at any point of a smooth

𝑚-manifold

the inverse

(𝑑𝑝 𝑥)− : R𝑚 → 𝑀𝑝 31

equals

𝑚.

Moreover,

3.3.

STRUCTURE OF TANGENT SPACES

of the operator

𝑑𝑝 𝑥

is defined by

(𝑑𝑝 𝑥)− (ℎ)(𝑓 ) = 𝜕ℎ (𝑓/𝑥 )(𝑥(𝑝)), for every fined at

ℎ

belonging to

R𝑚

and every smooth function

𝑓

locally de-

𝑝.

Proof.

Indeed, linearity appears completely obvious. Injectivity follows from the observation that, if

(𝑑𝑝 𝑥)(𝑣) = 𝑣𝑥 = 𝑤𝑥 = (𝑑𝑝 𝑥)(𝑤), then

𝑣(𝑓 ) = 𝜕𝑣(𝑥) (𝑓/𝑥 )(𝑥(𝑝)) = = 𝜕𝑤(𝑥) (𝑓/𝑥 )(𝑥(𝑝)) = = 𝑤(𝑓 ), for every 𝑓 . Hence we infer 𝑣 = 𝑤. For surjectivity, let 𝑣* be an 𝑚-vector, then the functional defined by 𝑣(𝑓 ) := 𝜕𝑣* (𝑓/𝑥 )(𝑥(𝑝)), satisfy the following equality

(𝑑𝑝 𝑥)(𝑣) = 𝑣* , as we claimed.

32

Chapter 4 Tangent frames 4.1 Geometric interpretation As we have seen, tangent spaces on smooth manifolds reveals, as in the Euclidean 𝑚-spaces 𝐸𝑚 , 𝑚 dimensional vector spaces attached to any point of 𝑝 of 𝑀 . Moreover, in Euclidean spaces, we define a tangent vector at 𝑝 as a vector 𝑣 with foot-point placed at the point 𝑝, that is as a pointed vector (𝑝, 𝑣). Then, we define the action of such pointed vector on usual differentiable functions 𝑓 by

(𝑝, 𝑣)(𝑓 ) = 𝜕𝑣 𝑓 (𝑝). Notice that, we can recover the vector 𝑣 from the way the pointed vector (𝑝, 𝑣) acts on differentiable functions, namely, by the following equality 𝑣 = ((𝑝, 𝑣)(e𝑗 ))𝑚 𝑗=1 , where e𝑗 represents the 𝑗 -th functional of the canonical dual basis e* , constituted by the Cartesian projections of the Euclidean 𝑚-space. We can so define a natural isomorphism 𝑥′𝑝 , depending on a chosen local 33

4.1.

GEOMETRIC INTERPRETATION

chart 𝑥, of a the tangent space 𝑀𝑝 onto the tangent space 𝑇𝑥(𝑝) R, by

𝑥′𝑝 : 𝑣 ↦→ (𝑥(𝑝), (𝑑𝑝 𝑥)(𝑣)), as we shall see in general, the above isomorphism is called the derivative of the local chart 𝑥.

Example.

If we consider R𝑚 as a differentiable manifold, the point partial derivative 𝜕𝑖 |𝑝 - sending each function 𝑓 , locally defined about 𝑝, into its partial derivative 𝜕𝑖 𝑓 (𝑝) - is a tangent vector at 𝑝. Geometrically, it corresponds to the point vector (𝑝, e𝑖 ), where e𝑖 stands for the 𝑖-th vector of the standard basis e. If e* represents the dual basis of e (basis of the dual of the Euclidean space), that is the system (e𝑖 )𝑚 𝑖=1 of canonical Cartesian projections, we obtain

𝜕𝑗 |𝑝 e𝑖 = 𝛿 𝑖 𝑗 = e𝑖 (e𝑗 ), for each index pair 𝑖, 𝑗 .

Remark.

In the definition of tangent vector, even in more direct way, we can write 𝑣(𝑓 ) = (𝜕𝑣(𝑥)|𝑥(𝑝) )(𝑓 /𝑥), which shows explicitly that the tangent vectors act like directional derivatives calculated at a point. Or, finally, we can write

𝑣(𝑓 ) = (𝑥(𝑝), 𝑣(𝑥))(𝑓 /𝑥), which shows explicitly how tangent vectors represent the counterpart of pointed vectors (𝑦, 𝑤) of Euclidean spaces. 34

CHAPTER 4.

TANGENT FRAMES

In the Euclidean space, we know the canonical e, we can transfer this basis by the inverse of the linear isomorphism 𝑑𝑝 𝑥. Observe that the tangent vector 𝑋𝑖 = (𝑑𝑝 𝑥)− (e𝑖 ), acts as it follows

𝑋𝑖 (𝑓 ) = 𝜕𝑖 (𝑓 /𝑥)(𝑥(𝑝)), for every 𝑓 . The preceding tangent vectors will be the subject of the following sections.

4.2 Tangent vectors determined by charts Consider a smooth real manifold 𝑀 with dimension 𝑚 and one its local coordinate system

𝑥 : 𝑈 → 𝑥(𝑈 ) centered at a point 𝑝 of 𝑀 .1 Then, for every index 𝑖 of the coordinate system 𝑥, we can define a tangent vector at 𝑝 ⃒ )︂ (︂ )︂ ⃒ (︂ ⃒ 𝜕 𝜕 𝜕 ⃒⃒ ⃒ : 𝐶𝑝∞ (𝑀, R) → R |𝑝 = = ⃒ ⃒ 𝜕𝑥 𝜕𝑥 𝜕𝑥 𝑖 𝑝,𝑖 𝑖 𝑝 by derivation of smooth function with respect to the coordinate system 𝑥 as it follows (︂ )︂ (︂ )︂ 𝜕 𝜕𝑓 (𝑝)(𝑓 ) := (𝑝) := 𝜕𝑖 (𝑓 /𝑥)(𝑥(𝑝)), 𝜕𝑥 𝑖 𝜕𝑥 𝑖 1A

local coordinate system on 𝑀 centered at 𝑝 is a function defined on an open part 𝑈 of 𝑀 containing 𝑝. If 𝑥 : 𝑈 → 𝑥(𝑈 )

is one such coordinate system, 𝑥 maps any point of its domain to the system of its components 𝑥 : 𝑞 ↦→ 𝑥(𝑞) = (𝑥𝑖 (𝑞))𝑚 𝑖=1 .

We denote by 𝑥𝑖 the 𝑖-th component of the function 𝑥.

35

4.2.

TANGENT VECTORS DETERMINED BY CHARTS

for every smooth real function 𝑓 locally defined on 𝑀 about the point 𝑝. Here we recall that

∙ the symbol 𝜕𝑖 stands for the usual 𝑖-th partial derivation on R𝑚 ; ∙ the function

𝑓 /𝑥 = 𝑓 ∘ 𝑥−

stands for the local representation of 𝑓 by the coordinate system 𝑥, defined from the image 𝑥(dom𝑓 ) to the real line R by

(𝑓 /𝑥)(𝑥(𝑝)) = 𝑓 (𝑝), for every point 𝑝 belonging to the domain of 𝑓 and also to the domain of the chart 𝑥.

Dangerous band.

Note that the single coordinate functions of the coordinate system 𝑥 do not intervene in the above definition. The above definition depends upon the entire coordinate system 𝑥 and on the index 𝑖 previously selected. Clearly, the just defined functional

𝑣 = 𝜕/𝜕𝑥𝑖 |𝑝 appears a bona fide tangent vector at 𝑝, because it acts like a directional derivative on representations 𝑓 /𝑥:

𝑣(𝑓 ) = 𝜕𝑒𝑖 (𝑓 /𝑥)(𝑥(𝑝)), where 𝑒𝑖 stands for the 𝑖-th vector of the standard basis in the Euclidean 𝑚-space. So, we can propose the following definition. 36

CHAPTER 4.

TANGENT FRAMES

Definition (of tangent vectors determined by a local coordinate system at a point). Under the previous conditions, we call the functional

(︂

𝜕 𝜕𝑥

)︂

(𝑝) : 𝐶𝑝∞ (𝑀, R) → R : 𝑓 ↦→ 𝜕𝑖 (𝑓 /𝑥)(𝑥(𝑝)),

𝑖

the 𝑖-th tangent vector determined by the coordinate system 𝑥 at the point 𝑝.

We can denote such tangent vector also by the various notations

𝜕 𝜕 𝜕 (𝑝) , |𝑝 , (𝑝)𝑖 , 𝜕/𝜕𝑥𝑖 (𝑝) , (𝜕/𝜕𝑥)𝑖 (𝑝) , (𝜕/𝜕𝑥)(𝑝)𝑖 . 𝜕𝑥 𝑖 𝜕𝑥 𝑖 𝜕𝑥 We emphasize that, fixed a coordinate system 𝑥, the 𝑖-th coordinate of 𝑥 is the function 𝑥𝑖 , while the 𝑖-th tangent vector induced by 𝑥 is the tangent vector (𝜕/𝜕𝑥)𝑖 |𝑝 , the first object shows the index 𝑖 in contravariant position, the second one shows the index 𝑖 in covariant position.

4.3 Tangent frames induced by local charts By the following notations

𝜕 𝜕 (𝑝) , |𝑝 , (𝜕/𝜕𝑥)(𝑝) , 𝜕/𝜕𝑥|𝑝 , 𝜕𝑥 𝜕𝑥 we denote the below family of tangent vectors

𝜕/𝜕𝑥|𝑝 = (𝜕/𝜕𝑥(𝑝)𝑖 )𝑚 𝑖=1 . 37

4.3.

TANGENT FRAMES INDUCED BY LOCAL CHARTS

Definition (tangent basis determined by a local coordinate system at a point). Under the previous conditions, we call the family (𝜕/𝜕𝑥)(𝑝) = ((𝜕/𝜕𝑥)𝑖 |𝑝 )𝑚 𝑖=1 . the tangent basis determined by the coordinate system 𝑥 at the point 𝑝.

The name of tangent basis is justified by the following theorem.

Theorem.

Let

𝑥

𝑀 around a point 𝑝. space 𝑀𝑝 can be uniquely

be a chart of a manifold

Then, any tangent vector

𝑣

in the tangent

written as a linear combination

𝑣=

∑︁

𝑣(𝑥)(𝜕/𝜕𝑥)𝑝 ,

of the family

(𝜕/𝜕𝑥)𝑝 , where the coefficient system

𝑣(𝑥)

remains defined by

𝑣(𝑥)𝑖 = 𝑣(𝑥𝑖 ), for every index

𝑖

of

𝑥.

Thus,

𝑀𝑝

is an

𝑚-dimensional

vector space

with basis

(𝜕/𝜕𝑥)𝑝 and the representation of the tangent vector

𝑣

in the coordinate system

𝑥, becomes the coordinate system of the tangent (𝜕/𝜕𝑥)𝑝 , that is 𝑣(𝑥) = [𝑣 | (𝜕/𝜕𝑥)𝑝 ], where by

vector

𝑣

in the basis

[𝑣|𝑏] we denote the coordinate system of a vector 𝑣 in a basis 𝑏.

38

CHAPTER 4.

TANGENT FRAMES

Proof.

We can easily see that each tangent vector at a point equals a unique linear combination of the family of tangent vectors determined by any local coordinate system at that point, that is:

𝑣(𝑓 ) = 𝜕𝑣(𝑥) (𝑓 /𝑥)(𝑥(𝑝)) = ∑︁ = 𝑣(𝑥)𝜕(𝑓 /𝑥)(𝑥(𝑝)) = ∑︁ = 𝑣(𝑥)(𝜕/𝜕𝑥)𝑝 (𝑓 ), for every 𝑓 . Concluding we can write: ∑︁ 𝑣= 𝑣(𝑥)(𝜕/𝜕𝑥)𝑝 , where the vector 𝑣(𝑥), i.e., the representation of the tangent vector 𝑣 in the coordinate system 𝑥, becomes the coordinate system of the tangent vector 𝑣 in the basis (𝜕/𝜕𝑥)𝑝 :

𝑣(𝑥) = [𝑣|(𝜕/𝜕𝑥)𝑝 ]. As we claimed.

4.4 Change of local frames In this section, we study the change of tangent frames. Let 𝑥 be a charts on a smooth manifold 𝑀 with domain 𝑈 and dimension 𝑚. We shall consider:

∙ the tangent frame induced by the chart 𝑥 at the point 𝑝, 𝜕/𝜕𝑥|𝑝 = ((𝜕/𝜕𝑥)𝑖 |𝑝 )𝑚 𝑖=1 , for every point 𝑝 of the open domain 𝑈 of 𝑥; 39

4.4.

CHANGE OF LOCAL FRAMES

∙ the (tangent) frame field induced by the chart 𝑥 upon its domain 𝑈, 𝜕/𝜕𝑥 : 𝑈 → 𝑇𝑈 (𝑀 )𝑚 : 𝑝 ↦→ 𝜕/𝜕𝑥|𝑝 , where

𝑇𝑈 (𝑀 ) =

⋃︁

𝑀𝑝

𝑝∈𝑈

represents the collection of all tangent vectors at 𝑝, with 𝑝 in 𝑈 (tangent bundle of 𝑀 upon 𝑈 );

∙ the vector field (𝜕/𝜕𝑥)𝑖 : 𝑈 → 𝑇𝑈 (𝑀 ) : 𝑝 ↦→ (𝜕/𝜕𝑥)𝑖 |𝑝 , for every index 𝑖 of the chart 𝑥.

Proposition (Transition matrix for local frames). (𝑈, 𝑥)

and

(𝑉, 𝑦)

Suppose

represent two coordinate charts on a manifold

𝑀.

Then, we see

(𝜕/𝜕𝑥)𝑖 |𝑝 =

𝑚 ∑︁

(𝜕𝑦 𝑗 /𝜕𝑥)𝑖 |𝑝 (𝜕/𝜕𝑦)𝑗 |𝑝 ,

𝑗=1

for every

𝑝

in

𝑈 ∩𝑉.

Consequently, we obtain

(𝜕/𝜕𝑥)𝑖 =

𝑚 ∑︁

(𝜕𝑦 𝑗 /𝜕𝑥)𝑖 (𝜕/𝜕𝑦)𝑗

𝑗=1

on the open intersection

𝑈 ∩𝑉 .

In other terms, by using the coordinate

brackets, we can write

[(𝜕/𝜕𝑥)𝑖 |𝑝 | (𝜕/𝜕𝑦)|𝑝 ] = (𝜕𝑦/𝜕𝑥)𝑖 |𝑝 , 40

CHAPTER 4.

for every

𝑝

in

𝑈 ∩𝑉

and every index

𝑖

of

TANGENT FRAMES

𝑥.

Proof.

At each point 𝑝 of the intersection 𝑈 ∩ 𝑉 , the families 𝜕/𝜕𝑥|𝑝 and 𝜕/𝜕𝑦|𝑝 are both bases for the tangent space 𝑇𝑝 𝑀 , so there exists a matrix field 𝑎 : 𝑈 ∩ 𝑉 → R𝑚,𝑚 defined by

𝑎(𝑝) = [𝑎𝑗 𝑖 (𝑝)]𝑚 𝑗,𝑖=1

such that

(𝜕/𝜕𝑥)𝑖 =

𝑚 ∑︁

𝑎𝑘𝑖 (𝜕/𝜕𝑦)𝑘

𝑘=1

on 𝑈 ∩ 𝑉 , that is,

(𝜕/𝜕𝑥)𝑖 |𝑝 =

𝑚 ∑︁

𝑎𝑘𝑖 (𝑝)(𝜕/𝜕𝑦)𝑘 |𝑝

𝑘=1

for every 𝑝 in 𝑈 ∩ 𝑉 . Applying both sides of the preceding equation to the component 𝑦 𝑗 of the coordinate system 𝑦 , we get 𝑗

(𝜕𝑦 /𝜕𝑥)𝑖 =

𝑚 ∑︁

𝑎𝑘𝑖 (𝜕𝑦 𝑗 /𝜕𝑦)𝑘 =

𝑘=1

= =

𝑚 ∑︁ 𝑘=1 𝑎𝑗 𝑖 ,

on 𝑈 ∩ 𝑉 , as we claimed.

41

𝑎𝑘𝑖 𝛿 𝑗𝑘 =

4.4.

CHANGE OF LOCAL FRAMES

42

Part II Contravariant and covariant vectors

43

Chapter 5 Contravariant vectors 5.1 Introduction In the preceding sections, we have already associated a contravariant vector 𝑣 ¯ with any tangent vector 𝑣 at a point 𝑝 of an 𝑚-dimensional smooth manifold 𝑀 ; namely, the function

𝑣¯ : 𝒜𝑝 → R𝑚 sending any local coordinate system 𝑥 at 𝑝 to the representation 𝑣(𝑥) of the tangent vector 𝑣 in the coordinate system 𝑥, that is

𝑣¯(𝑥) := 𝑣𝑥 = (𝑣(𝑥𝑖 ))𝑚 𝑖=1 , for every coordinate system 𝑥 at 𝑝. It turns out that the above concept opens the doors to the general classic concept of contravariant vector, in the form introduced by Gregorio Ricci-Curbastro and Tullio Levi-Civita. 45

5.2.

DEFINITION

5.2 Definition Let us formalize that important definition of contravariant vector at a point.

Definition (of contravariant vector). 𝑚

smooth manifold

𝑀

with atlas

𝒜,

a point

Consider a dimension

𝑝

of the manifold

𝑀

and

the coordinate system collection

𝒜𝑝 = {𝑦 ∈ 𝒜 : 𝑝 ∈ dom(𝑦)}, 𝑀 at 𝑝. Then, a contravariant vector upon 𝑀 at the point 𝑝, say 𝑣 , is a mapping

of all the charts of

𝑣 : 𝒜𝑝 → R𝑚 such that, for every pair of charts

𝑥

and

𝑦

of

𝒜𝑝 ,

we observe

𝑣(𝑦) = 𝑑𝑥(𝑝) (𝑦/𝑥 )(𝑣(𝑥)), where

𝑦/𝑥

denotes the transition mapping from the chart

𝑥

to

𝑦.

In terms of the matrix associated with the differential of the transition map, we can write:

𝑣(𝑦) = 𝐽𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥)), where

𝐽𝑥(𝑝) (𝑦/𝑥) = (𝜕𝑗 (𝑦 𝑖 /𝑥)|𝑥(𝑝) )𝑚 𝑖,𝑗=1 = = (𝜕𝑦/𝜕𝑥)|𝑝 represents the Jacobian matrix of the transition mapping 𝑦/𝑥 calculated at the point 𝑥(𝑝). 46

CHAPTER 5.

CONTRAVARIANT VECTORS

5.3 Contravariant vector components From the definition of contravariant vector in matrix form, we can see

𝑣 𝑖 (𝑦) = [𝐽𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥))]𝑖 = 𝑚 ∑︁ = 𝜕𝑗 (𝑦 𝑖 /𝑥)|𝑥(𝑝) [𝑣(𝑥)]𝑗 = 𝑗=1

=

𝑚 ∑︁

(𝜕𝑦 𝑖 /𝜕𝑥)𝑗 |𝑝 𝑣 𝑗 (𝑥) =

𝑗=1

= (𝜕𝑦 𝑖 /𝜕𝑥)|𝑝 · 𝑣(𝑥), which provides the contravariance definition in terms of components, where we wrote 𝑣 𝑖 (𝑦) for the component 𝑣(𝑦)𝑖 of the vector 𝑣(𝑦). We call the mapping

𝑣 𝑖 : 𝒜𝑝 → R : 𝑥 ↦→ 𝑣 𝑖 (𝑥) = 𝑣(𝑥)𝑖 , sending every of chart 𝑥 to the 𝑖-th component 𝑣(𝑥)𝑖 , the 𝑖-th component of the contravariant vector 𝑣 . With any tangent vector 𝑣 at 𝑝, we associate the contravariant vector 𝑣¯ defined by 𝑣¯𝑖 (𝑥) = 𝑣(𝑥𝑖 ), for every index 𝑖 and for every chart 𝑥 at 𝑝. Or, equivalently,

𝑣¯(𝑥) = 𝑣𝑥 , for every chart 𝑥 at 𝑝, where 𝑣𝑥 stands for the representation of 𝑣 at 𝑥. 47

5.4.

CONTRAVARIANT VECTOR SPACE

5.4 Contravariant vector space We denote the collection of all contravariant vectors at 𝑝 by

𝑇01 (𝑀, 𝑝). It reveals immediately a vector space with respect to the natural pointwise operations. We shall see that this vector space is isomorphic with the tangent space 𝑀𝑝 .

Theorem.

Consider, the mapping

⃗ : 𝑀𝑝 → 𝑇 1 (𝑀, 𝑝) : 𝑣 ↦→ ⃗𝑣 , (.) 0 𝑝 sending any tangent vector

𝑣

at

𝑝

into its contravariant vector

⃗𝑣 : 𝒜𝑝 → R𝑚 : 𝑥 ↦→ 𝑣(𝑥) = 𝑣𝑥 . Then, the mapping

Proof.

⃗ (.) 𝑝

reveals linear, injective and surjective.

Linearity reveals straightforward.

Injectivity.

⃗𝑣 = 𝑤, ⃗ then, for every local chart 𝑥 at 𝑝, we get

𝑣𝑥 = ⃗𝑣 (𝑥) = 𝑤(𝑥) ⃗ = 𝑤𝑥 , thus

𝑣(𝑓 ) = 𝜕𝑣𝑥 (𝑓 /𝑥)(𝑥(𝑝)) = = 𝜕𝑤𝑥 (𝑓 /𝑥)(𝑥(𝑝)) = = 𝑤(𝑓 ), 48

If

CHAPTER 5.


for each smooth test function 𝑓 .

Surjectivity.

On the other hand, if 𝑣 represents a contravariant vector at 𝑝, then it can define a directional derivative functional

𝜕𝑣 : 𝐶𝑝∞ (𝑀 ) → R belonging to the tangent space 𝑀𝑝 , by

𝜕𝑣 (𝑓 ) := 𝑑𝑥(𝑝) (𝑓 /𝑥)(𝑣(𝑥)), for every test function 𝑓 and by fixing arbitrarily a chart 𝑥 at 𝑝; the above definition does not depend on 𝑥 because

𝑑𝑦(𝑝) (𝑓 /𝑦)(𝑣(𝑦)) = = = =

𝑑𝑦(𝑝) (𝑓 /𝑦)(𝑑𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥))) = (𝑑𝑦(𝑝) (𝑓 /𝑦) ∘ 𝑑𝑥(𝑝) (𝑦/𝑥))(𝑣(𝑥)) = (𝑑𝑥(𝑝) ((𝑓 /𝑦) ∘ (𝑦/𝑥))(𝑣(𝑥)) = 𝑑𝑥(𝑝) (𝑓 /𝑥)(𝑣(𝑥)),

for any other chart 𝑦 at 𝑝. Clearly, we can conclude that

𝜕⃗𝑣 = 𝑣, for every contravariant vector 𝑣 , and hence the mapping

⃗ : 𝑀𝑝 → 𝑇 1 (𝑀, 𝑝) (.) 0 𝑝 admits a section (right inverse)

𝜕|𝑝 : 𝑇01 (𝑀, 𝑝) → 𝑀𝑝 : 𝑣 ↦→ 𝜕𝑣 , and, therefore, reveals also surjective. 49

5.5.

CONTRAVARIANT 2-TENSORS

Concluding, the mapping

⃗ : 𝑀𝑝 → 𝑇 1 (𝑀, 𝑝) : 𝑣 ↦→ ⃗𝑣 , (.) 0 𝑝 sending any tangent vector 𝑣 at 𝑝 into its contravariant vector ⃗𝑣 reveals a linear isomorphism of vector spaces, whose inverse is the mapping

𝜕|𝑝 : 𝑇01 (𝑀, 𝑝) → 𝑀𝑝 : 𝑣 ↦→ 𝜕𝑣 , defined above. As we claimed.

5.5 Contravariant 2-tensors In this section, we offer the classic (and rigorous) definition of contravariant 2-tensor in matrix form, component form and using the scalar product of matrices, along the classic approach of Einstein’s General Relativity papers.

Definition (contravariant 2-tensor). smooth manifold

𝑀

with atlas

𝒜,

a point

𝑝

Consider a dimension

of the manifold

𝑀

𝑚

and the

coordinate system collection

𝒜𝑝 = {𝑦 ∈ 𝒜 : 𝑝 ∈ dom(𝑦)}, of all the charts of

𝑀

𝑀 at the point 𝑝,

𝑝. Then, a contravariant 2-tensor upon 𝑣 , is a mapping

at

say

𝑣 : 𝒜𝑝 → R𝑚,𝑚 such that, for every pair of charts

𝑥

and

𝑦

of

𝒜𝑝 ,

we verify

𝑚 ∑︁ 𝑚 ∑︁ 𝑣 (𝑦) = (𝜕𝑦 𝑖 /𝜕𝑥)ℎ |𝑝 (𝜕𝑦 𝑗 /𝜕𝑥)𝑘 |𝑝 [𝑣(𝑥)]ℎ𝑘 = 𝑖𝑗

ℎ=1 𝑘=1 𝑖

= [(𝜕𝑦 /𝜕𝑥)|𝑝 ⊗ (𝜕𝑦 𝑗 /𝜕𝑥)|𝑝 ] · 𝑣(𝑥), 50

CHAPTER 5.


which provides the contravariance definition in terms of components, 𝑖𝑗 𝑖𝑗 where we wrote 𝑣 (𝑦) for the component 𝑣(𝑦) of the matrix 𝑣(𝑦).

Remark.

Explicitly, in the above definition, we have

𝑣 𝑖𝑗 (𝑦) = [𝐽𝑥(𝑝) (𝑦/𝑥)𝑣(𝑥)𝑡 𝐽𝑥(𝑝) (𝑦/𝑥)]𝑖𝑗 = 𝑚 ∑︁ 𝑚 ∑︁ = 𝜕ℎ (𝑦 𝑖 /𝑥)|𝑥(𝑝) 𝜕𝑘 (𝑦 𝑗 /𝑥)|𝑥(𝑝) [𝑣(𝑥)]ℎ𝑘 , ℎ=1 𝑘=1

for each pair of charts 𝑥, 𝑦 at 𝑝.

5.6 Tensor product Let us consider now two contravariant vectors 𝑣, 𝑤 at 𝑝. We desire to calculate the natural product of such two vectors, in order to obtain a double contravariant tensor

𝑢 = 𝑣 ⊗ 𝑤. Define 𝑢 as a contravariant 2-tensor upon 𝑀 at the point 𝑝, a mapping 𝑣 ⊗ 𝑤 : 𝒜𝑝 → R𝑚,𝑚 such that, for every chart 𝑥 of 𝒜𝑝 , we see

(𝑣 ⊗ 𝑤)(𝑥) := 𝑣(𝑥) ⊗ 𝑤(𝑥), which goes to mean

𝑢(𝑥)𝑖𝑗 := 𝑣(𝑥)𝑖 𝑤(𝑥)𝑗 ,

for every pair of indexes 𝑖, 𝑗 . 51

5.6.

TENSOR PRODUCT

Theorem. Proof.

The mapping

𝑣⊗𝑤

reveals a contravariant 2-tensor.

For, we observe that

(𝑣 ⊗ 𝑤)(𝑦)𝑖𝑗 = 𝑣 𝑖 (𝑦)𝑤𝑗 (𝑦) = 𝑚 𝑚 ∑︁ ∑︁ 𝑖 ℎ = (𝜕𝑦 /𝜕𝑥)ℎ |𝑝 [𝑣(𝑥)] (𝜕𝑦 𝑗 /𝜕𝑥)𝑘 |𝑝 [𝑤(𝑥)]𝑘 = ℎ=1

=

𝑚 𝑚 ∑︁ ∑︁

𝑘=1

(𝜕𝑦 𝑖 /𝜕𝑥)ℎ |𝑝 (𝜕𝑦 𝑗 /𝜕𝑥)𝑘 |𝑝 [𝑣(𝑥)]ℎ [𝑤(𝑥)]𝑘 =

ℎ=1 𝑘=1 𝑖

= [(𝜕𝑦 /𝜕𝑥)|𝑝 ⊗ (𝜕𝑦 𝑗 /𝜕𝑥)|𝑝 ] · (𝑣(𝑥) ⊗ 𝑤(𝑥)) = = [(𝜕𝑦 𝑖 /𝜕𝑥)|𝑝 ⊗ (𝜕𝑦 𝑗 /𝜕𝑥)|𝑝 ] · (𝑣 ⊗ 𝑤)(𝑥), for any pair 𝑥, 𝑦 of charts.

52

Chapter 6 Covariant vectors 6.1 Covariant vectors Let us introduce the classic definition of covariant vector.

Definition (of covariant vector). smooth manifold

𝑀

with atlas

𝒜,

Consider an

𝑝

a point

𝑚-dimensional 𝑀 and the

of the manifold

coordinate system collection

𝒜𝑝 = {𝑦 ∈ 𝒜 : 𝑝 ∈ dom(𝑦)}, of all the charts of

𝑀

at

𝑝.

Then,

a covariant vector upon 𝑀 at

the point 𝑝, say 𝑤 , is a mapping

𝑤 : 𝒜𝑝 → R𝑚 such that, for every pair of charts

𝑥

and

𝑦

of

𝒜𝑝 ,

we obtain

𝑤(𝑦) = 𝑑𝑦(𝑝) (𝑥/𝑦)* (𝑤(𝑥)), where

𝑦

𝑥/𝑦

denotes the transition mapping from the coordinate system

to the coordinate system

𝑥

and where

𝑑𝑦(𝑝) (𝑥/𝑦)* 53

6.1.

COVARIANT VECTORS

represents the adjoint operator of the differential

𝑑𝑦(𝑝) (𝑥/𝑦), with respect to the standard scalar product in

R𝑚 .

Consider a covariant vector 𝑤 centered at a point 𝑝 of an 𝑚differential manifold 𝑀 with atlas 𝒜. For every index 𝑖, we define the 𝑖-th component of the covariant vector 𝑤, as the application

𝑤𝑖 : 𝒜𝑝 → R : 𝑥 ↦→ 𝑤(𝑥)𝑖 , whose variation (action) is determined by

𝑤𝑖 (𝑦) =

𝑚 ∑︁

(𝜕𝑥𝑗 /𝜕𝑦)𝑖 |𝑝 𝑤𝑗 (𝑥) =

𝑗=1

= = = =

(𝜕𝑥/𝜕𝑦)𝑖 |𝑝 · 𝑤(𝑥) = [𝑡 𝐽𝑦(𝑝) (𝑥/𝑦)(𝑤(𝑥))]𝑖 = [𝑡 (𝜕𝑥/𝜕𝑦)𝑝 (𝑤(𝑥))]𝑖 = [𝑑𝑦(𝑝) (𝑥/𝑦)* (𝑤(𝑥))]𝑖 ,

for every couple of coordinate systems 𝑥, 𝑦 . Recall that

𝑑𝑦(𝑝) (𝑥/𝑦)* represents the adjoint operator of the differential

𝑑𝑦(𝑝) (𝑥/𝑦), with respect to the standard scalar product in R𝑚 , defined by (︀ )︀ (︀ )︀ 𝑑𝑦(𝑝) (𝑥/𝑦)* (𝑘) | ℎ = 𝑘 | 𝑑𝑦(𝑝) (𝑥/𝑦)(ℎ) , for every couple of vectors ℎ, 𝑘 ∈ R𝑚 . Recall, moreover, that the adjoint operator of a certain linear operator 𝐿 is canonically associated with the transpose matrix of 𝐿. 54

CHAPTER 6.

COVARIANT VECTORS

6.2 The gradient of a real function In this section, we shall consider the gradient at a point of a real smooth function, locally defined over a smooth manifold, with respect to a local coordinate chart around that point. We shall show that the mapping sending each chart 𝑥 of the manifold, around a point 𝑝, into the gradient at 𝑝 of a function 𝑓 , with respect to the chart 𝑥 itself, reveals a covariant vector defined on the manifold at the point 𝑝.

Definition (gradient of a smooth function with respect to a local coordinates chart). Let 𝑀 be a smooth manifold and let 𝑓 be a smooth function locally defined upon each coordinate system

𝑥

around

𝑝,

𝑀,

at a point

𝑝

of

𝑀.

For

we can define the point-gradient

grad𝑥 (𝑓 )(𝑝) = (𝜕𝑓 /𝜕𝑥)𝑝 := [(𝜕𝑓 /𝜕𝑥)𝑖 |𝑝 ]𝑚 𝑖=1 , of the smooth real function

𝑓 : 𝑉 → R, with respect to

𝑥.

We can now prove immediately the basic theorem of this section.

Theorem.

Consider the below vector function

charts around a point

𝑝

of an

𝑚-differential

𝑤

- defined on local

manifold

𝑀

with atlas

𝒜

- defined by

𝑤 : 𝒜𝑝 → R𝑚 : 𝑥 ↦→ (𝜕𝑓 /𝜕𝑥)𝑝 . The mapping

𝑤

associates with any coordinate system

grad𝑥 (𝑓 )(𝑝) = (𝜕𝑓 /𝜕𝑥)𝑝 , 55

𝑥

the gradient

6.3.

INVARIANT SCALARS

of a certain smooth real function

𝑓 : 𝑉 → R, locally defined at the point

𝑝,

reveals a covariant vector at

Proof.

with respect to

𝑥.

Then, the mapping

𝑤

𝑝.

Indeed, for any couple of charts 𝑥, 𝑦 about 𝑝, we know

(𝜕𝑓 /𝜕𝑥)𝑖 |𝑝 =

𝑚 ∑︁ (𝜕𝑦 𝑗 /𝜕𝑥)𝑖 |𝑝 (𝜕𝑓 /𝜕𝑦)𝑗 |𝑝 = 𝑗=1

= (𝜕𝑦/𝜕𝑥)𝑖 |𝑝 · (𝜕𝑓 /𝜕𝑦)|𝑝 , that is

𝑤𝑖 (𝑥) = (𝜕𝑦/𝜕𝑥)𝑖 |𝑝 · 𝑤(𝑦), as we claimed.

6.3 Invariant scalars We desire to calculate the pointwise Euclidean scalar product of a contravariant vector 𝑣 times a covariant vector 𝑤, that is the application

(𝑣|𝑤) : 𝒜𝑝 → R : 𝑥 ↦→ (𝑣(𝑥)|𝑤(𝑥)). We shall see that this pointwise scalar product reveals a scalar invariant at 𝑝, that is a constant mapping from 𝒜𝑝 to R. Our proof essentially reproduces a classic Einstein’s proof from his first paper about General Relativity.

Theorem.

Consider a differential manifold

a contravariant vector

𝑣

at

𝑝

𝑀,

a point

and a covariant vector

point. Then, the application

(𝑣|𝑤) : 𝒜𝑝 → R : 𝑥 ↦→ (𝑣(𝑥)|𝑤(𝑥)), 56

𝑤

𝑝

of

𝑀,

at the same

CHAPTER 6.

COVARIANT VECTORS

pointwise Euclidean scalar product of the two above vectors, reveals a scalar invariant at

𝑝,

that is a constant mapping from

Proof 1 (Einstein’s proof). (𝑣(𝑦)|𝑤(𝑦)) = =

𝑚 ∑︁

𝒜𝑝

to

R.

For, we observe that

𝑣 𝑖 (𝑦)𝑤𝑖 (𝑦) =

𝑖=1 𝑚 ∑︁

[𝐽𝑥(𝑝) (𝑦/𝑥)(𝑣(𝑥))]𝑖 [𝑡 𝐽𝑦(𝑝) (𝑥/𝑦)(𝑤(𝑥))]𝑖 =

𝑖=1 𝑚 ∑︁ 𝑚 𝑚 ∑︁ ∑︁ (𝜕𝑥𝑘 /𝜕𝑦)𝑖 |𝑝 𝑤𝑘 (𝑥) = = (𝜕𝑦 𝑖 /𝜕𝑥)𝑗 |𝑝 𝑣 𝑗 (𝑥)

= = = =

𝑖=1 𝑗=1 𝑚 ∑︁ 𝑚 ∑︁ 𝑚 ∑︁

𝑘=1

(𝜕𝑦 𝑖 /𝜕𝑥)𝑗 |𝑝 (𝜕𝑥𝑘 /𝜕𝑦)𝑖 |𝑝 𝑣 𝑗 (𝑥)𝑤𝑘 (𝑥) =

𝑘=1 𝑗=1 𝑖=1 𝑚 𝑚 ∑︁ ∑︁

(𝜕𝑥/𝜕𝑦|𝑝 . 𝜕𝑦/𝜕𝑥|𝑝 )𝑘𝑗 𝑣 𝑗 (𝑥)𝑤𝑘 (𝑥) =

𝑘=1 𝑗=1 𝑚 𝑚 ∑︁ ∑︁

𝛿 𝑘𝑗 𝑣 𝑗 (𝑥)𝑤𝑘 (𝑥) =

𝑘=1 𝑗=1 𝑚 ∑︁ 𝑗

𝑣 (𝑥)𝑤𝑗 (𝑥) =

𝑗=1

= (𝑣(𝑥)|𝑤(𝑥)), as we claimed. In the above proof, we used the following property.

Property.

The row-column product

𝜕𝑥/𝜕𝑦|𝑝 . 𝜕𝑦/𝜕𝑥|𝑝 , 57

6.3.

INVARIANT SCALARS

of the Jacobian matrices associated with the two mutually inverse transition maps

𝑥/𝑦

and

𝑦/𝑥,

is the identity

(𝑚, 𝑚)

matrix

𝛿.

Proof.

Indeed, the two above Jacobian matrices correspond to the following mutually inverse linear operators:

𝑑𝑦(𝑝) (𝑥/𝑦) , 𝑑𝑥(𝑝) (𝑦/𝑥), as we can easily see by the following application of the chain rule:

𝑑𝑦(𝑝) (𝑥/𝑦) ∘ 𝑑𝑥(𝑝) (𝑦/𝑥) = 𝑑𝑥(𝑝) (𝑥/𝑦 ∘ 𝑦/𝑥) = = 𝑑𝑥(𝑝) (I𝑥(𝑈 ∩𝑉 ) ) = = IR𝑚 , where 𝑈 is the domain of the chart 𝑥, 𝑉 is the domain of the chart 𝑦 and I𝑥(𝑈 ∩𝑉 ) is the identity mapping on 𝑥(𝑈 ∩ 𝑉 ).

Second proof of the theorem. (𝑣(𝑦)|𝑤(𝑦)) = = = =

Alternatively, we see at once

(𝐽𝑥(𝑝) (𝑦/𝑥)𝑣(𝑥)|𝑡 𝐽𝑦(𝑝) (𝑥/𝑦)𝑤(𝑥)) = (𝐽𝑦(𝑝) (𝑥/𝑦).𝐽𝑥(𝑝) (𝑦/𝑥)𝑣(𝑥)|𝑤(𝑥)) = (𝛿𝑣(𝑥)|𝑤(𝑥)) = (𝑣(𝑥)|𝑤(𝑥)),

by profiting of the behavior of transpose matrices in scalar products.

Third proof of the theorem. (𝑣(𝑦)|𝑤(𝑦)) = = = =

Alternatively, we see again at once

(𝑑𝑥(𝑝) (𝑦/𝑥)𝑣(𝑥)|𝑑𝑦(𝑝) (𝑥/𝑦)* 𝑤(𝑥)) = (𝑑𝑦(𝑝) (𝑥/𝑦) ∘ 𝑑𝑥(𝑝) (𝑦/𝑥)𝑣(𝑥)|𝑤(𝑥)) = (IR𝑚 𝑣(𝑥)|𝑤(𝑥)) = (𝑣(𝑥)|𝑤(𝑥)), 58

CHAPTER 6.

COVARIANT VECTORS

by profiting of the behavior of adjoint operators in scalar products. So, we conclude the proof. So we can define the scalar product of a contravariant and a covariant vector as the unique value of their pointwise scalar product. In such a way, the covariant vectors acts as linear functionals on 𝑇01 (𝑀, 𝑝) as the contravariant vectors acts as linear functionals on 𝑇10 (𝑀, 𝑝).

59

6.3.

INVARIANT SCALARS

60

Part III Derivation

61

Chapter 7 Derivatives 7.1 Introduction Consider a differentiable function

𝑓 : R𝑚 → R𝑛 between Euclidean spaces and a point 𝑝 of its domain R𝑚 . In finite dimensional vector calculus, the Jacobian matrix of 𝑓 at the point 𝑝, defined by

𝐽𝑓 (𝑝) := (𝜕𝑖 𝑓 𝑗 (𝑝))𝑛,𝑚 𝑗,𝑖=1 , reveals the (𝑛, 𝑚)-matrix associated with the derivative 𝑓 ′ (𝑝) of the function 𝑓 at the point 𝑝. The derivative of the function 𝑓 at the point 𝑝 is the linear operator defined as follows:

𝑓𝑝′ : R𝑚 → R𝑛 : 𝑣 ↦→ 𝜕𝑣 (𝑓 )(𝑝), where 𝜕𝑣 (𝑓 )(𝑝) denotes the usual directional derivative, with respect to 𝑣 , of the function 𝑓 , at the point 𝑝. 63

7.1.

INTRODUCTION

It appears, therefore, quite natural, when seeking an analogous derivative concept for a map

𝑓 :𝑀 →𝑁 between manifolds 𝑀 and 𝑁 , to look again for a linear transformation. Previously, we defined a vector space at each point of a manifold, locally representing the manifold itself, this suggests to define a linear transformation 𝑓𝑝′ = 𝑓*𝑝 = 𝑇𝑝 𝑓 : 𝑀𝑝 → 𝑁𝑓 (𝑝) , between the respective tangent spaces, locally representing the function 𝑓 near the point 𝑝. We would like that the linear operator 𝑓𝑝′ correspond to the matrix 𝐽𝑓 (𝑝), when the manifold 𝑀 equals the Euclidean space R𝑚 and 𝑁 equals R𝑛 and whenever the tangent space R𝑘𝑞 , of any Euclidean 𝑘 space at a point 𝑞 is identified with the vector space of all pointed vectors (𝑞, 𝑣), with 𝑣 in R𝑘 . In other terms, we require that

𝑓𝑝′ (𝑝, 𝑣) = (𝑓 (𝑝), 𝐽𝑓 (𝑝)𝑣) for all vector 𝑣 in R𝑚 . Now, if

𝑔 : R𝑛 → R is differentiable, then by the Chain Rule,

𝑓𝑝′ (𝑝, 𝑣)(𝑔) = = = = = = =

(𝑓 (𝑝), 𝐽𝑓 (𝑝)𝑣)(𝑔) = (𝜕𝐽𝑓 (𝑝)𝑣 𝑔)(𝑓 (𝑝)) = 𝑑𝑔(𝑓 (𝑝))(𝑑𝑓 (𝑝)𝑣) = (𝑑𝑔(𝑓 (𝑝)) ∘ 𝑑𝑓 (𝑝))(𝑣) = 𝑑(𝑔 ∘ 𝑓 )(𝑝)(𝑣) = 𝜕𝑣 (𝑔 ∘ 𝑓 )(𝑝) = (𝑝, 𝑣)(𝑔 ∘ 𝑓 ). 64

CHAPTER 7.

DERIVATIVES

7.2 Definition of derivatives The above argumentations motivate the following definition.

Definition (of derivative).

and

𝑁

denote two smooth

𝑚 and 𝑛 respectively. Consider 𝑈 of 𝑝 in 𝑀 and a map

manifolds of dimensions

𝑀,

𝑀

Let

a point

𝑝

of

at

𝑝

an open neighborhood

𝑓 :𝑈 →𝑁 differentiable at the point

𝑝. 1

Then, we define the derivative of

𝑓

as the map

𝑓𝑝′ : 𝑀𝑝 → 𝑁𝑓 (𝑝) given by

𝑓𝑝′ (𝑣)(𝑔) := 𝑣(𝑔 ∘ 𝑓 ), where

𝑔

lies in the vector fibered space

𝐶𝑓∞(𝑝) (𝑁 ), of smooth functions locally defined on in the tangent space 1A

𝑁

at the point

𝑓 (𝑝),

and

𝑣

lies

𝑀𝑝 .

map between two differentiable manifolds 𝑓 :𝑈 →𝑁

is called differentiable at a point 𝑝 of 𝑈 if, for each real smooth function 𝑔 : 𝑉 → R,

locally defined around 𝑓 (𝑝), the composition 𝑔 ∘ 𝑓 is smooth at the point 𝑝. In other terms, 𝑓 is called differentiable at the point 𝑝 if any composition 𝑔 ∘ 𝑓 lies in 𝐶𝑝∞ (𝑈 ), as soon as the function 𝑔 lives in 𝐶𝑓∞(𝑝) (𝑁 ).

65

7.3.

LOCAL JACOBIAN MATRICES

It is clear from the definition that 𝑓𝑝′ is a linear transformation. We shall avoid to denote the derivative 𝑓𝑝′ by the symbol 𝑓 ′ (𝑝). We shall reserve the latter symbol for the classic Frechet derivative, when it applies.

7.3 Local Jacobian matrices In this section we examine the local representation of derivatives via Jacobian matrices.

Proposition.

With notation as in the above Definition, let

coordinate map around the point

𝑝

of the open part

𝑈, 𝑦

𝑥

be a

a coordinate

map around the point 𝑓 (𝑝) in the manifold 𝑁 . Then, the matrix of the ′ derivative 𝑓𝑝 , with respect to the induced bases 𝜕/𝜕𝑥|𝑝 and 𝜕/𝜕𝑦|𝑓 (𝑝) is the Jacobian matrix of the

(𝑥, 𝑦)-representative

of the function

𝑓,

i.e.,

the mapping 𝑦 𝑓/𝑥

at the point

:= 𝑦 ∘ 𝑓 ∘ 𝑥− ,

𝑥(𝑝).

Proof. The 𝑗 -th component of a tangent vector 𝑤 at a point 𝑞 of the open domain dom(𝑦), in the basis 𝜕/𝜕𝑦|𝑞 is the real number

𝑤𝑗 (𝑦) = 𝑤(𝑦 𝑗 ). Therefore, we only need to calculate the image of the basis

𝑋 = 𝜕/𝜕𝑥|𝑝 by the tangent map 𝑓𝑝′ , obtaining a family of tangent vectors at 𝑓 (𝑝), and then, we need to apply the family 𝑓𝑝′ (𝑋) to the coordinate function 𝑦 𝑗 to get the 𝑗 -th row of the desired associated matrix. In other terms, 66

CHAPTER 7.

DERIVATIVES

the matrix of the derivative 𝑓𝑝′ , with respect to the bases 𝑋 = 𝜕/𝜕𝑥|𝑝 and 𝑌 = 𝜕/𝜕𝑦|𝑓 (𝑝) is the matrix

[𝑓𝑝′ ]𝑌𝑋 = [𝑓𝑝′ (𝑋)|𝑌 ], whose 𝑗 -th row is the vector

[𝑓𝑝′ (𝑋)|𝑌 ]𝑗 = ([𝑓𝑝′ (𝑋𝑖 )|𝑌 ]𝑗 )𝑚 𝑖=1 , or, equivalently, whose 𝑖-th column is the system of coordinates

[𝑓𝑝′ (𝑋𝑖 )|𝑌 ] = 𝑓𝑝′ (𝑋𝑖 )(𝑦). Therefore, the (𝑗, 𝑖) element of the matrix [𝑓𝑝′ ]𝑌𝑋 is

([𝑓𝑝′ ]𝑌𝑋 )𝑗 𝑖 = [𝑓𝑝′ (𝑋𝑖 )|𝑌 ]𝑗 = = 𝑓𝑝′ (𝑋𝑖 )(𝑦)𝑗 = = 𝑓𝑝′ (𝑋𝑖 )(𝑦 𝑗 ). Moreover, we immediately see that

𝑓𝑝′ (𝜕/𝜕𝑥𝑖 |𝑝 )(𝑦 𝑗 ) = (𝜕/𝜕𝑥)𝑖 |𝑝 (𝑦 𝑗 ∘ 𝑓 ) = = 𝜕𝑖 (𝑦 𝑗 ∘ 𝑓 ∘ 𝑥− )(𝑥(𝑝)) = = (𝐽𝑥(𝑝) (𝑦 ∘ 𝑓 ∘ 𝑥− ))𝑗 𝑖 , for every index 𝑗 and 𝑖. Concluding, we can write

[𝑓𝑝′ ]𝑌𝑋 = 𝐽𝑥(𝑝) (𝑦 𝑓/𝑥 ), as we claimed. 67

7.4.

BASIC PROPERTIES OF THE TANGENT MAP

7.4 Basic properties of the tangent map Let us begin with an elementary example regarding the identity mapping of a manifold.

Example.

It follows easily - from the definition of tangent map that the identity map I𝑀 , of the manifold 𝑀 , shows as its derivative at a point 𝑝 of 𝑀 the identity map I𝑀𝑝 of the tangent space 𝑀𝑝 of the manifold 𝑀 at the point 𝑝. More generally, we can proof the following theorem.

Theorem.

Let

𝑈

be an open subset of a smooth manifold

𝑀.

The

differential of the immersion

𝑗 : 𝑈 → 𝑀 : 𝑢 ↦→ 𝑢, at any point

𝑝

of

𝑈,

is the identity map

I𝑀𝑝 : 𝑀𝑝 → 𝑀𝑝 , of the tangent space

Proof.

𝑀𝑝

to the smooth manifold

𝑀

at the point

𝑝.

We can easily see that

𝑗𝑝′ (𝑣)(𝑓 ) = 𝑣(𝑓 ∘ 𝑗) = = 𝑣(𝑓 |𝑈 ) = = 𝑣(𝑓 ), for every tangent vector to 𝑀 at 𝑝 and every smooth real function 𝑓 locally defined at the point 𝑝. We used the property affirming that the value of a tangent vector at 𝑝 on a smooth function depends only on the germ of that function at the point 𝑝. 68

Chapter 8 Chain rules and diffeomorphisms 8.1 Chain rule Now we can prove the most important property of the tangency: the chain rule.

Theorem (chain rule). 𝑓, 𝑔

Let consider two differentiable mappings

between manifolds:

𝑓 : 𝑀 → 𝑁, 𝑔 : 𝑁 → 𝑃. Then, the composition

𝑔∘𝑓 is differentiable, and

(𝑔 ∘ 𝑓 )′𝑝 = 𝑔𝑓′ (𝑝) ∘ 𝑓𝑝′ , or, equivalently

(𝑔 ∘ 𝑓 )*𝑝 = 𝑔*𝑓 (𝑝) ∘ 𝑓*𝑝 . 69

8.2.

DIFFEOMORPHISMS AND DERIVATIVES

Proof. Let 𝑤 belong to the tangent space 𝑇𝑝 𝑁 and let ℎ be a smooth local function at 𝑔(𝑓 (𝑝)) of the manifold 𝑃 . Then (𝑔 ∘ 𝑓 )′𝑝 (𝑤)(ℎ) = 𝑤(ℎ ∘ (𝑔 ∘ 𝑓 )) and

(𝑔𝑓′ (𝑝) ∘ 𝑓𝑝′ )(𝑤)(ℎ) = 𝑔𝑓′ (𝑝) (𝑓𝑝′ (𝑤))(ℎ) = = (𝑓𝑝′ (𝑤))(ℎ ∘ 𝑔) = = 𝑤(ℎ ∘ 𝑔 ∘ 𝑓 ), as we claimed.

8.2 Diffeomorphisms and derivatives One particular case of the Chain Rule, at once, offers us the following property about diffeomorphisms. A diffeomorphism between two manifolds is an invertible mapping, between the two manifolds, which is smooth together with its inverse.

Theorem.

Consider a diffeomorphism

𝑓 :𝑀 →𝑁 between two manifolds. Then, at any point

𝑀,

𝑝

of the domain manifold

the derivative

𝑓𝑝′ : 𝑀𝑝 → 𝑁𝑓 (𝑝) , reveals an isomorphism of vector spaces. Moreover, the inverse of that − derivative is the derivative of the inverse mapping 𝑓 at the point 𝑓 (𝑝), that is,

(𝑓𝑝′ )− = (𝑓 − )′𝑓 (𝑝) . 70

CHAPTER 8.

CHAIN RULES AND DIFFEOMORPHISMS

Proof. The function 𝑓 is a diffeomorphism between 𝑀 and 𝑁 if there exists a differentiable mapping 𝑔:𝑁 →𝑀 such that

𝑔 ∘ 𝑓 = I𝑀 and

𝑓 ∘ 𝑔 = I𝑁 . By the chain rule, we immediately get

I𝑇𝑝 𝑀 = (I𝑀 )′𝑝 = = (𝑔 ∘ 𝑓 )′𝑝 = = 𝑔𝑓′ (𝑝) ∘ 𝑓𝑝′ , and, setting 𝑞 := 𝑓 (𝑝) (so that 𝑝 = 𝑔(𝑞)), we obtain

𝑓𝑝′ ∘ 𝑔𝑓′ (𝑝) = = = =

′ 𝑓𝑔(𝑞) ∘ 𝑔𝑞′ = (𝑓 ∘ 𝑔)′𝑞 = (I𝑁 )′𝑞 = I𝑇𝑞 𝑁 .

Hence, the derivatives 𝑓𝑝′ and 𝑔𝑞′ are vector space isomorphisms. On the other hand, the diffeomorphism 𝑔 equals the inverse of 𝑓 ,

𝑔 = 𝑓 −, and, from

𝑓𝑝′ ∘ 𝑔𝑞′ = I𝑇𝑞 𝑁 , 71

8.3.

TANGENT BASES UNDER DERIVATIVES

we infer

(𝑓𝑝′ )− = (𝑔)′𝑞 = = (𝑓 − )′𝑓 (𝑝) , as we claimed. .

Corollary (Invariance of dimension under diffeomorphism). If an open subset

𝑈

phic to an open set

𝑛

equals dimension

𝑛-space reveals 𝑚-space R𝑚 , then

of the real Euclidean

𝑉 of 𝑚.

the Euclidean

diffeomordimension

Proof. Let 𝑓 : 𝑈 → 𝑉 be a diffeomorphism of the manifold 𝑈 to the manifold 𝑉 and let 𝑝 be a point of 𝑈 . By the above corollary, the derivative 𝑓𝑝′ : 𝑈𝑝 → 𝑉𝑓 (𝑝) , of the diffeomorphism 𝑓 at the point 𝑝, reveals an isomorphism of vector spaces. Since the tangent space 𝑈𝑝 is isomorphic to the Euclidean space R𝑛 and the tangent space 𝑉𝑓 (𝑝) reveals isomorphic to the Euclidean space R𝑚 , we infer that 𝑛 = 𝑚.

8.3 Tangent bases under derivatives We denote here by 𝑟 the identity chart on R𝑚 , and, if (𝑈, 𝑥) is a chart about a point 𝑝 of a manifold 𝑀 of dimension 𝑚, we set

𝑥𝑖 = 𝑟𝑖 ∘ 𝑥. Since

𝑥 : 𝑈 → 𝑥(𝑈 ) is a diffeomorphism onto its image, then the differential

𝑥′𝑝 : 𝑀𝑝 → 𝑥(𝑈 )𝑞 , 72

CHAPTER 8.

CHAIN RULES AND DIFFEOMORPHISMS

where 𝑞 = 𝑥(𝑝), is a vector space isomorphism. In particular, the tangent space 𝑀𝑝 shows the same dimension 𝑚 as the manifold 𝑀 , as we already know. We propose the following interesting property about the transformation of tangent bases under derivatives.

Proposition. 𝑀.

Let

(𝑈, 𝑥)

be a chart about a point

𝑝

in a manifold

Then

𝑥′𝑝 (𝜕/𝜕𝑥𝑖 |𝑝 ) = 𝜕𝑖 |𝑥(𝑝) , for every index

Proof.

𝑖

of

𝑥.

For any smooth function 𝑓 locally defined at 𝑞 = 𝑥(𝑝),

𝑥′𝑝 (𝜕/𝜕𝑥𝑖 |𝑝 )(𝑓 ) = (𝜕/𝜕𝑥𝑖 |𝑝 )(𝑓 ∘ 𝑥) = = 𝜕𝑖 |𝑞 (𝑓 ∘ 𝑥 ∘ 𝑥− ) = = 𝜕𝑖 |𝑥(𝑝) (𝑓 ), as we claimed. .

73

8.3.

TANGENT BASES UNDER DERIVATIVES

74

Part IV Paths on manifolds

75

Chapter 9 Paths on manifolds 9.1 Paths on a Manifold Definition (smooth path).

A

smooth path in a manifold 𝑀 is by

definition a smooth map

𝑝 : 𝐼 → 𝑀, from some open interval of the real line interval

𝐼

will be called the

element of

𝐼

will be named

𝐼

into the manifold

𝑀.

The

parameter space of the path 𝑝 and each parameter of the path.

When the real number 0 belongs to the parameter domain 𝐼 of a path 𝑝 : 𝐼 → 𝑀, we say that the path 𝑝 is a path originating at the point 𝑝0 of 𝑀 if the path 𝑝 passes through the point 𝑝0 at the parameter 0, which means

𝑝(0) = 𝑝0 . 77

9.1.

PATHS ON A MANIFOLD

Definition (vector derivative of a path). path in a manifold

𝑀,

Consider a smooth

a smooth map

𝑝: 𝐼 →𝑀 from some open interval of the real line

𝐼

into the manifold

𝑀.

Let

𝑦

represent the identity mapping of the interval 𝐼 . The vector derivative 𝑝′ (𝑦0 ) of the path 𝑝 at the point 𝑦0 ∈ 𝐼 is defined as the tangent vector

(︂

′

𝑝 (𝑦0 ) := 𝑝*𝑦0 of the tangent space

𝑑⎮ ⎮ ⎮ 𝑑𝑦 𝑦0

)︂

𝑇𝑝(𝑦0 ) 𝑀 .

In the conditions of the above definition, the tangent vector (︂ ⃒ )︂ 𝑑⃒ ′ 𝑝 (𝑦0 ) := 𝑝*𝑦0 ⃒ , 𝑑𝑦 𝑦0 acts as it follows

𝑝′ (𝑦0 ) : 𝑓 ↦→ (𝑑/𝑑𝑦)(𝑓 ∘ 𝑝)(𝑦0 ) for every real smooth function 𝑓 on 𝑀 , where

(𝑑/𝑑𝑦)(𝑓 ∘ 𝑝) = 𝑑(𝑓 ∘ 𝑝)/𝑑𝑦 represents the classic derivative of the real function

𝑓 ∘ 𝑝 : 𝐼 → R, and

𝑑/𝑑𝑦|𝑦0 : 𝐶𝑦∞0 (𝐼) → R

is the derivative functional on the interval 𝐼 at the parameter value 𝑦0 . 78

CHAPTER 9.

PATHS ON MANIFOLDS

For what concerns the action of the tangent derivative on the local charts, we readily obtain

𝑝′ (𝑦0 )(𝑥) = 𝐷(𝑥 ∘ 𝑝)(𝑦0 ) ∈ R𝑚 , for every local coordinate system 𝑥 on the manifold 𝑀 , where 𝐷(𝑥 ∘ 𝑝) represents the classic derivative of the path 𝑥 ∘ 𝑝.

Remark. If the path 𝑝 reveals injective, we also say that the vector derivative 𝑝′ (𝑦0 ) is the vector derivative of the path 𝑝 at the point 𝑝(𝑦0 ). Alternative notations for the vector derivative 𝑝′ (𝑦0 ) are

𝑑𝑝 (𝑦0 ) 𝑑𝑦 and

𝑑⎮ ⎮ ⎮ 𝑝, 𝑑𝑦 𝑦0 where 𝑦 represents the identity function on the interval 𝐼 .

9.2 Derivatives by re-parametrizations Definition (vector derivative of a path with respect to a reparametrization). Consider a smooth path in a manifold 𝑀 , a smooth map

𝑝: 𝑌 →𝑀 from some open interval of the real line

𝑧

𝑌

into the manifold

represent a local coordinate system of the interval

derivative

(𝑑𝑝/𝑑𝑧)(𝑦0 ) 79

𝑌.

The

𝑀.

Let

vector

9.3.

TANGENT VECTOR VS CALCULUS DERIVATIVE

of the path

𝑧

𝑝

at the point

𝑦0 ∈ 𝐼 ,

with respect to the coordinate system

is defined as the tangent vector

(︂ (𝑑𝑝/𝑑𝑧)(𝑦0 ) := 𝑝*𝑦0 of the tangent space

𝑑⎮ ⎮ ⎮ 𝑑𝑧 𝑦0

)︂

𝑇𝑝(𝑦0 ) 𝑀 .

In the conditions of the above definition, the tangent vector (︂ ⎮ )︂ 𝑑⎮ (𝑑𝑝/𝑑𝑧)(𝑦0 ) := 𝑝*𝑦0 ⎮ , 𝑑𝑧 𝑦0 acts as it follows

𝑥 ↦→ 𝐷(𝑥 ∘ 𝑝 ∘ 𝑧 − )(𝑧(𝑦0 ))

for every coordinate system 𝑥 on 𝑀 , where

𝐷(𝑥 ∘ 𝑝 ∘ 𝑧 − ) represents the classic derivative of the path

𝑥 ∘ 𝑝 ∘ 𝑧 − : 𝑌 → 𝑥(𝑈 ).

9.3 Tangent vector vs calculus derivative When

𝑝 : 𝐼 → R𝑛

represents a smooth path in the Euclidean space R𝑛 , the notation 𝑝′ (𝑦0 ) for the vector derivative could generate some confusion, because, in Calculus, 𝑝′ (𝑦0 ) represents, instead, the 𝑛-vector whose components are the slopes of the graphs of the single components 𝑝𝑗 at the point 𝑦0 , which are real numbers. Therefore, in Calculus, 𝑝′ (𝑦0 ) is an 𝑛-vector 80

CHAPTER 9.

PATHS ON MANIFOLDS

and not a derivation at the point 𝑝(𝑦0 ). Let 𝑥 be the standard coordinate system on the target space R𝑛 (i.e., the identity function I of the space itself). By our definition, 𝑝′ (𝑦0 ) is a tangent vector at the point 𝑝(𝑦0 ), hence it is a linear combination of the point derivation basis

𝜕/𝜕𝑥|𝑝(𝑦0 ) . On the other hand, in Calculus notation 𝑝′ (𝑦0 ) is the vector derivative of a vector-valued function and it is therefore a simple 𝑛-vector. To distinguish between these two meanings of the derivative 𝑝′ (𝑦0 ), when 𝑝 maps an open interval 𝐼 into the Euclidean space R𝑛 , we will write

𝑝(𝑦 ˙ 0) for the classic Calculus derivative.

Example (Vector derivative versus calculus derivative).

Let 𝑥 represent the identity function of the real line and let

𝑝:𝐼→R represent a path in the real line R. We see immediately that

𝑝′ (𝑦0 ) = 𝑝(𝑦 ˙ 0 ) (𝑑/𝑑𝑥)|𝑝(𝑦0 ) , for every 𝑦0 in 𝐼 .

81

9.4.

AN ELEMENTARY EXAMPLE

9.4 An elementary example . Let 𝑚, 𝑛 represent two natural numbers. Define the path

𝑐 : R → R2 by

𝑐(𝑝) = (𝑝𝑚 , 𝑝𝑛 ), for every real number 𝑝. Then, the point derivation 𝑐′ (𝑝) is a linear combination of the two canonical point derivations 𝜕1 |𝑞 and 𝜕2 |𝑞 at the point 𝑞 := 𝑐(𝑝) of the path 𝑐, so that

𝑐′ (𝑝) = 𝑎𝜕1 |𝑞 + 𝑏𝜕2 |𝑞 . To compute the coefficient 𝑎, we evaluate the above linear combination on the first canonical projection 𝑥 := e1 , obtaining

𝑎 = = = = = =

(𝑎𝜕1 |𝑞 + 𝑏𝜕2 |𝑞 )𝑥 = 𝑐′ (𝑝)(𝑥) = 𝑐*𝑝 (𝑑/𝑑𝑧|𝑝 )(𝑥) = 𝑑/𝑑𝑧|𝑝 (𝑥 ∘ 𝑐) = 𝑑/𝑑𝑧|𝑝 (𝑧 𝑚 ) = 𝑚𝑝𝑚−1

where 𝑧 represents the identity function on the real line. Similarly, we evaluate the above linear combination on the second canonical projection 𝑦 := e2 , obtaining 82

CHAPTER 9.

𝑏 = = = = = =

PATHS ON MANIFOLDS

(𝑎𝜕1 |𝑞 + 𝑏𝜕2 |𝑞 )𝑦 = 𝑐′ (𝑝)(𝑦) = 𝑐*𝑝 (𝑑/𝑑𝑧|𝑝 )(𝑦) = 𝑑/𝑑𝑧|𝑝 (𝑦 ∘ 𝑐) = 𝑑/𝑑𝑧|𝑝 (𝑧 𝑛 ) = 𝑛𝑝𝑛−1 .

Thus,

𝑐′ (𝑝) = 𝑎𝜕1 |𝑞 + 𝑏𝜕2 |𝑞 = = (𝑚𝑝𝑚−1 )𝜕1 |𝑞 + (𝑛𝑝𝑛−1 )𝜕2 |𝑞 . In terms of the standard basis 𝜕|𝑐(𝑝) of the tangent space 𝑇𝑐(𝑝) (R2 ), we can write [︂ ]︂ 𝑚𝑝𝑚−1 ′ [𝑐 (𝑝)|𝜕(𝑞)] = . 𝑛𝑝𝑛−1

9.5 Vector derivative in local coordinates In general, to compute the vector derivative of a smooth path 𝑝 in the Euclidean space R𝑚 , we simply differentiate the components of 𝑝 with respect to the identity chart 𝑟. The following proposition shows that our definition of the tangent vector derivative of a path on a smooth manifold agrees with the usual definition in vector calculus.

Proposition (Vector derivative of a path in local coordinates). Let 𝑝:𝐼→𝑀 83

9.5.

VECTOR DERIVATIVE IN LOCAL COORDINATES

𝑚-manifold 𝑀 , and let (𝑈, 𝑥) be a local point 𝑝0 = 𝑝(𝑞0 ) of 𝑀 , with 𝑞0 in the interval

be a smooth path in a smooth coordinate chart about a

𝐼.

Set

𝑝𝑖𝑥 := 𝑥𝑖 ∘ 𝑝, 𝑖-th component of the ′ derivative 𝑝 (𝑞0 ) is given by for the

′

𝑝 (𝑞0 ) =

path

𝑚 ∑︁

𝑝

𝑝˙𝑖𝑥 (𝑞0 )

𝑖=1

in the chart

(︂

𝜕 𝜕𝑥

)︂ ⎮ ⎮ ⎮ 𝑖

𝑥.

Then, the vector

.

𝑝(𝑞0 )

Thus, relative to the tangent basis 𝜕/𝜕𝑥|𝑝0 of the tangent space ′ the vector derivative 𝑝 (𝑞0 ) is represented by the 𝑚-vector

[︀ of the Euclidean space Proof.

]︀ 𝑝˙1𝑥 (𝑞0 ), ..., 𝑝˙𝑚 𝑥 (𝑞0 ) ,

R𝑚 .

Straightforward.

84

𝑇𝑝0 𝑀 ,

Chapter 10 Paths and tangent vectors 10.1 Path initial tangent vector In a manifold 𝑀 , every smooth path 𝑝, passing through a point 𝑝0 , gives rise to a tangent vector 𝑝′ (𝑞0 ) belonging to the tangent space 𝑇𝑝0 𝑀 , where 𝑝(𝑞0 ) = 𝑝0 . Conversely, we can show that every tangent vector 𝑣 in 𝑀𝑝0 is the vector derivative of some path 𝑝 passing through the point 𝑝0 , at some parameter 𝑞0 such that

𝑝(𝑞0 ) = 𝑝0 , as it follows.

Proposition (Existence of a path with a given initial derivative vector). For any point 𝑝0 of a smooth 𝑚-manifold 𝑀 and any tangent vector

𝐼,

𝑣

in the tangent space

𝑀𝑝0 ,

there exist an open interval

of the real line, centered at 0 and a smooth path

𝑝:𝐼→𝑀 such that

𝑝(0) = 𝑝0

and

𝑝′ (0) = 𝑣. 85

10.1.

PATH INITIAL TANGENT VECTOR

Proof.

Let (𝑈, 𝑥) be a chart centered at 𝑝0 , i.e., assume that 𝑥(𝑝0 ) equals the origin 0 of R𝑚 . We know that

𝑣=

𝑚 ∑︁

𝑣𝑥𝑖 (𝜕/𝜕𝑥)𝑖 |𝑝0 ,

𝑖=1

at the point 𝑝0 , for the convenient uniquely determined coefficient system 𝑣𝑥 . Let 𝑟 be the standard coordinate system on R𝑚 (i.e., the identity function of R𝑚 , whose associated coordinate family is the dual of the canonical basis e). Then, the single coordinate functions of the chart 𝑥 appear defined by

𝑥𝑖 = 𝑟𝑖 ∘ 𝑥, for every index 𝑖 of the chart 𝑥. To find a smooth path 𝑝 passing through 𝑝0 and with 𝑝′ (0) = 𝑣 , we use a path

𝑞 : R → R𝑚 , in R𝑚 , with 𝑞(0) = 0 and ′

𝑞 (0) =

𝑚 ∑︁

𝑣𝑥𝑖 (𝜕/𝜕𝑟)𝑖 (0).

𝑖=1

We shall send the path 𝑞 to the manifold 𝑀 via the local parametrization 𝑥− . First of all, note that such an elementary path 𝑞 , in the 𝑚-Euclidean space, can be defined as it follows

𝑞(𝑧0 ) = 𝑧0 𝑣𝑥 , for every 𝑧0 in a convenient open interval 𝐼 of the real line; the interval 𝐼 needs to be chosen sufficiently small so that the point 𝑞(𝑧0 ) lies in 86

CHAPTER 10.

PATHS AND TANGENT VECTORS

the neighborhood 𝑥(𝑈 ) of the origin 0, for every 𝑧0 in 𝐼 . Define the required path 𝑝 as the below composition

𝑝 := 𝑥− ∘ 𝑝 : 𝐼 → 𝑈. Then, we obtain

𝑝(0) = 𝑥− (𝑞(0)) = 𝑥− (0) = 𝑝0 , and, by the chain rule,

𝑝′ (0) = (𝑥− ∘ 𝑞)′ (0) = (︂ ⎮ )︂ 𝑑⎮ − ′ = (𝑥 ∘ 𝑞)0 ⎮ = 𝑑𝑦 0 (︂ ⎮ )︂ 𝑑⎮ − ′ ′ = ((𝑥 )𝑝0 ∘ 𝑞0 ) ⎮ = 𝑑𝑦 0 (︂ (︂ ⎮ )︂)︂ 𝑑⎮ − ′ = (𝑥 )𝑝0 𝑞0′ = ⎮ 𝑑𝑦 0 )︃ (︃ 𝑚 ∑︁ (︂ 𝜕 )︂ ⎮ ⎮ = (𝑥− )′𝑝0 𝑣𝑥𝑖 ⎮ = 𝜕𝑟 0 𝑖 𝑖=1 )︂ (︂(︂ 𝑚 ⎮ )︂ ∑︁ 𝜕 ⎮ 𝑖 − ′ = 𝑣𝑥 (𝑥 )𝑝0 ⎮ = 𝜕𝑟 0 𝑖 𝑖=1 (︂ )︂ ⃒ 𝑚 ∑︁ 𝜕 ⃒ 𝑖 = 𝑣𝑥 ⃒ = 𝜕𝑥 𝑖 𝑝0 𝑖=1 = 𝑣, as we claimed.

10.2 Tangent vectors as vector derivatives We defined a tangent vector at a point 𝑝 of a manifold as a point directional derivative functional at 𝑝. Using paths, we can now interpret 87

10.2.

TANGENT VECTORS AS VECTOR DERIVATIVES

a tangent vector geometrically as a vector derivative of a smooth path.

Proposition. ifold

𝑀

Suppose

𝑣

is a tangent vector at a point

and consider any real function

𝑓

𝑝0

of a man∞ in the fibered space 𝐶𝑝0 (𝑀 ).

If

𝑝:𝐼→𝑀 𝑝0

is a smooth path originating at

with

𝑝′ (0) = 𝑣, then

𝑑⎮ ⎮ ⎮ (𝑓 ∘ 𝑝), 𝑑𝑦 0 chart on 𝐼 .

𝑣(𝑓 ) = where

𝑦

represents the identity

Proof.

By the definitions of the tangent derivative 𝑝′ (0) and recalling that the derivative 𝑝* : 𝑦0 ↦→ 𝑝′𝑦0 acts as below

𝑝′𝑦0 (𝑤)(𝑓 ) = 𝑤(𝑓 ∘ 𝑝),

for every tangent vector 𝑤 of the interval 𝐼 at the point 𝑦0 and every smooth function 𝑓 locally defined about the point 𝑝(𝑦0 ) of 𝑀 , we immediately get

𝑣(𝑓 ) = 𝑝′ (0)(𝑓 ) = (︂ ⃒ )︂ 𝑑⃒ ′ = 𝑝0 ⃒ (𝑓 ) = 𝑑𝑦 0 𝑑 ⃒⃒ = ⃒ (𝑓 ∘ 𝑝), 𝑑𝑦 0 for every smooth function 𝑓 locally defined about the point 𝑝(0) of 𝑀 , as we claimed. 88

CHAPTER 10.

PATHS AND TANGENT VECTORS

10.3 Derivatives and paths Let

𝑓 :𝑁 →𝑀 be a smooth map between two manifolds. We recall that, at each point 𝑝 of the manifold 𝑁 , the map 𝑓 induces a linear map between tangent spaces, called its derivative at 𝑝, denoted by

𝑓𝑝′ : 𝑁𝑝 → 𝑀𝑓 (𝑝) and defined as follows: if the tangent vector 𝑣 lives in the tangent space 𝑁𝑝 , then the derivative 𝑓𝑝′ (𝑣) is the tangent vector in the space 𝑀𝑓 (𝑝) described by 𝑓𝑝′ (𝑣)(𝑔) = 𝑣(𝑔 ∘ 𝑓 ), for every function 𝑓 belonging to 𝐶𝑓∞(𝑝) (𝑀 ). We have so introduced two ways of computing the derivative of a smooth map, in terms of derivations at a point (just its definition) and in terms of local coordinates (by the local Jacobian matrix). The next proposition gives another way of computing the derivative 𝑓𝑝′ , using paths.

Proposition.

Consider a smooth map

𝑓 :𝑁 →𝑀 between manifolds, a point

𝑣

in the tangent space

𝑝

𝑇𝑝 𝑁 .

of the manifold Let the map

𝑞:𝐼→𝑁 89

𝑁

and a tangent vector

10.3.

DERIVATIVES AND PATHS

represent a smooth path originating at the point with vector derivative

𝑣

at the point

map of the open interval

𝐼.

Proof.

that

𝑦

𝑁,

represent the identity

𝑑 ⃒⃒ ⃒ (𝑓 ∘ 𝑞). 𝑑𝑦 0

In other words, the tangent vector

𝑓 ∘𝑞

and let

in the manifold

Then

𝑓𝑝′ (𝑣) =

image path

𝑝

𝑝

𝑓𝑝′ (𝑣)

is the vector derivative of the

at the parameter point 0 (note that

𝑓 (𝑞(0)) = 𝑓 (𝑝)).

By hypothesis, 𝑞(0) = 𝑝 and 𝑞 ′ (0) = 𝑣 . Then, we easily see

𝑓𝑝′ (𝑣) = 𝑓𝑝′ (𝑞 ′ (0)) = = 𝑓𝑝′ (𝑞0′ (𝑑/𝑑𝑦|0 )) = (︂ ⎮ )︂ 𝑑⎮ ′ ′ = (𝑓𝑝 ∘ 𝑞0 ) ⎮ = 𝑑𝑦 0 (︂ ⎮ )︂ 𝑑⎮ ′ = (𝑓 ∘ 𝑞)0 ⎮ = 𝑑𝑦 0 𝑑⎮ ⎮ = ⎮ (𝑓 ∘ 𝑞), 𝑑𝑦 0 as we claimed.

90

Part V Covectors

91

Chapter 11 Covectors and differentials 11.1 Cotangent vectors In order to introduce the fundamental concept of a differential 1-form, we need to define the cotangent space to a point 𝑝 of a smooth manifold.

Definition (cotangent space).

Let

𝑀

be a smooth manifold and

𝑀 . The cotangent space of the manifold 𝑀 at the point 𝑝, denoted also by 𝑇𝑝* (𝑀 ) or 𝑇𝑝* 𝑀 , is defined as the algebraic * dual space 𝑀𝑝 of the tangent space 𝑀𝑝 of 𝑀 at the point 𝑝, 𝑝

a point of

𝑇𝑝* 𝑀 := (𝑀𝑝 )* = Hom(𝑀𝑝 , R). It is the vector space of all linear functionals from the tangent space

𝑀𝑝

(at

𝑝)

towards the real line.

The elements of the cotangent space at a point 𝑝 are called cotan-

gent vectors of the manifold at the point 𝑝. 93

11.2.

DIFFERENTIAL 1-FORMS

Definition (covector). is called also a covector at

An element of the cotangent space

𝑇𝑝* 𝑀

𝑝.

Thus, a covector 𝜔 at a point 𝑝 is a real linear functional

𝜔 : 𝑀𝑝 → R, over the tangent space at the point 𝑝.

Definition (cotangent bundle space).

By the symbol

𝑇 * (𝑀 ),

𝑀

(at any

we represent the collection of all covectors on a manifold point

𝑝

of the manifold

𝑀 ),

i.e., the below union

𝑇 * (𝑀 ) :=

⋃︁

𝑇𝑝* (𝑀 ).

𝑝∈𝑀

We call that collection by the name of

cotangent bundle space.

The cotangent bundle space is a fibered space with respect to the projection 𝜋 : 𝑇 * (𝑀 ) → 𝑀, defined by

𝜋(𝜔) = 𝑝 if 𝜔 ∈ 𝑀𝑝* , for every covector 𝜔 .

11.2 Differential 1-forms Now we can define the fundamental concept of differential form on a manifold. 94

CHAPTER 11.

COVECTORS AND DIFFERENTIALS

Definition (covector field).

A covector field, or differential 1-

form, or 1-form, on a smooth manifold

𝑀,

is a mapping

𝜔 : 𝑀 → 𝑇 * (𝑀 ) assigning to each point

𝑝

of

𝑀

a covector

𝜔𝑝

of

𝑀

at the point

𝑝.

In this sense, the covector fields appear as dual objects of vector fields on the manifold 𝑀 (the mappings which assign to each point 𝑝 of 𝑀 a tangent vector at 𝑝). Dual objects in the sense that we can define a pairing (scalar product)

(.|.) : (𝜔, 𝑣) ↦→ (𝜔|𝑣) := 𝜔(𝑣), where the action 𝜔(𝑣), of a 1-form on a vector field 𝑣 , is point-wise defined as the function

𝜔(𝑣) : 𝑝 ↦→ 𝜔𝑝 (𝑣𝑝 ), from 𝑀 to R. A vector field 𝑣 is defined smooth if the real function

𝑣(𝑓 ) : 𝑝 ↦→ 𝑣𝑝 (𝑓 ), reveals smooth, for every smooth function 𝑓 globally defined on 𝑀 . A differential form is defined smooth if the function 𝜔(𝑣) reveals smooth for every smooth vector field 𝑣 .

Note. We know many reasons motivating the introduction of differential forms in Differential Manifold Theory: first of all, differential forms represents many classes of physical quantities, in Newtonian, 95

11.3.

DIFFERENTIAL OF A FUNCTION

Lagrangian and Hamiltonian Mechanics, in thermodynamics, fluiddynamics and Einstein Relativity. Moreover, differential forms stay at the foundations of the Integration Theory in differential manifolds. Finally, we shall observe that we can pull back differential forms under smooth maps, in contrast to vector fields, as, in general, we cannot push forward a vector field under a smooth map.

11.3 Differential of a function Covector fields appear naturally from the study of local coordinate systems. Consider an open part 𝑉 of a manifold 𝑀 and let

𝑣 : 𝑉 → 𝑇𝑀 be a smooth vector field on the submanifold 𝑉 of 𝑀 . Fixed a chart

𝑥 : 𝑈 → 𝑥(𝑈 ), then, at each point 𝑝 of the domain of the chart 𝑥, we can expand the tangent vector 𝑣𝑝 in the (induced) local frame

𝜕/𝜕𝑥|𝑝 , as it follows

𝑣𝑝 =

𝑚 ∑︁

𝑣𝑝𝑖 (𝑥)(𝜕/𝜕𝑥)𝑖 |𝑝 ,

𝑖=1

where the coefficient 𝑣𝑝𝑖 (𝑥) represents the 𝑖-th component of the representation 𝑣𝑝 (𝑥) of the tangent vector 𝑣𝑝 in the local chart 𝑥. 96

CHAPTER 11.


The above coefficient 𝑣𝑝𝑖 (𝑥) clearly depends on the vector 𝑣𝑝 and on the chart 𝑥. The chart 𝑥 induces, in fact, a natural linear functional:

𝐿𝑖𝑥 (𝑝) : 𝑇𝑝 𝑀 → R : 𝑤 ↦→ 𝑤(𝑥)𝑖 , i.e., a covector at 𝑝. Moreover, as 𝑝 varies over the domain 𝑈 of the chart 𝑥, the chart 𝑥 induces the below covector field on 𝑈

𝐿𝑖𝑥 : 𝑈 → 𝑇 * 𝑀 : 𝑝 ↦→ 𝐿𝑖𝑥 (𝑝). We shall see that the above example of covector field belong to the important class of exact differential forms, for this scope we need the following basic definition.

Definition (differential).

We define the

differential at 𝑝 of a

smooth function

𝑓 : 𝑉 → R, locally defined on

𝑀

at

𝑝

as the following linear functional (covector)

𝑑𝑝 𝑓 : 𝑇𝑝 𝑀 → R : 𝑣 ↦→ 𝑣(𝑓 ). We define

the differential of 𝑓 as the covector field

𝑑𝑓 : 𝑉 → 𝑇 * 𝑀 sending each point

𝑞

of the domain

𝑉

to the form

𝑑𝑞 𝑓 ,

11.4 Exact 1-forms Definition (exact differentials).

A covector field

𝜔 : 𝑉 → 𝑇 *𝑀 97

defined above.

11.5.

DERIVATIVE VERSUS DIFFERENTIALS

is named

exact 1-form if

𝜔 = 𝑑𝑓 for some smooth function

𝑓 : 𝑉 → R.

For example, the above covector field 𝐿𝑖𝑥 reveals none other than the exact differential 1-form 𝑑𝑥𝑖 , differential of the 𝑖-th component of the chart 𝑥, selecting the 𝑖-th component of a vector field

𝑣 : 𝑈 → 𝑇 𝑀, relative to the frame field

𝜕/𝜕𝑥 : 𝑈 → (𝑇 𝑀 )𝑚 : 𝑝 ↦→ 𝜕/𝜕𝑥|𝑝 , induced by 𝑥 on its domain 𝑈 .

Proof.

Indeed, we immediately read:

(𝑑𝑥𝑖 )(𝑣𝑝 ) = 𝑣𝑝 (𝑥𝑖 ) = = 𝑣𝑝 (𝑥)𝑖 = = 𝑣𝑝𝑖 (𝑥) = = 𝐿𝑖𝑥 (𝑝)(𝑣𝑝 ), for every point 𝑝 of 𝑈 and every index 𝑖.

11.5 Derivative versus differentials Instead of 𝑑𝑝 𝑓 or (𝑑𝑓 )𝑝 , we also write 𝑑𝑓 |𝑝 , for the value of the differential 1-form 𝑑𝑓 at the point 𝑝. This is analogous to the notations for a tangent vector; for instance:

(𝑑/𝑑𝑦)𝑝 = 𝑑/𝑑𝑦|𝑝 , 98

CHAPTER 11.


where 𝑦 represents the identity function of an open subset 𝑌 of the real line, or a local coordinate system on it. Before, we encountered the notion of derivative, defined by

𝑓′ : 𝑀 →

⋃︁

Hom(𝑀𝑝 , 𝑁𝑓 (𝑝) ) : 𝑝 ↦→ 𝑓 ′ (𝑝),

𝑝∈𝑀

for a map 𝑓 between manifolds, where the linear operator 𝑓 ′ (𝑝) acts as it follows

𝑓𝑝′ (𝑣)(𝑔) = 𝑣(𝑔 ∘ 𝑓 ), for every tangent vector 𝑣 of 𝑀 at 𝑝 and for every real smooth function 𝑔 locally defined at the point 𝑓 (𝑝) of 𝑁 . Let us compare the two notions of derivative and differential.

Proposition.

𝑓 : 𝑀 → R is a smooth function, then, for every point 𝑝 of the domain manifold 𝑀 and every tangent vector 𝑣 of the tangent space 𝑀𝑝 , we obtain If

𝑓𝑝′ (𝑣) = (𝑑𝑓 )𝑝 (𝑣)(𝑑/𝑑𝑟)𝑓 (𝑝) , where

𝑟

represents the identity function of the real line.

Since the derivative 𝑓𝑝′ (𝑣) belongs to the tangent space 𝑇𝑓 (𝑝) R, then there exists a real number 𝑎 such that Proof.

𝑓𝑝′ (𝑣) = 𝑎(𝑑/𝑑𝑟)𝑓 (𝑝) . To evaluate the coefficient 𝑎, we apply the above derivative to the 99

11.6.

CRITICAL POINTS OF REAL FUNCTIONS

identity function 𝑟. We obtain:

𝑎 = = = = =

𝑎(𝑑/𝑑𝑟)𝑓 (𝑝) (𝑟) = 𝑓𝑝′ (𝑣)(𝑟) = 𝑣(𝑟 ∘ 𝑓 ) = 𝑣(𝑓 ) = (𝑑𝑓 )𝑝 (𝑣),

as we claimed. This proposition shows that - under the canonical identification of the tangent space 𝑇𝑓 (𝑝) R with the real line R, via the isomorphism (covector) 𝑑𝑝 𝑟 : 𝑎(𝑑/𝑑𝑟)𝑓 (𝑝) ↦→ 𝑎, induced by the identity function 𝑟 of the real line R, the derivative 𝑓𝑝′ corresponds to the differential 𝑑𝑝 𝑓 .

11.6 Critical points of real functions In terms of the differential 𝑑𝑓 , we can define the critical points of a real function defined on a manifold. We say that a point

𝑝

of a manifold

𝑀

is a

critical point of a

smooth real function

𝑓 :𝑀 →R if and only if

(𝑑𝑓 )𝑝 = 0, that is, if and only if the differential of tional on

𝑀𝑝 . 100

𝑓

at

𝑝

is the zero linear func-

CHAPTER 11.


We could show that local minimum points and local maximum points of a smooth function

𝑓 :𝑀 →R are critical points.

Indeed, let 𝑝 be an extreme point for 𝑓 , then any local representation 𝑓/𝑥 of the function 𝑓 - by some local chart 𝑥 at 𝑝 - shows an extreme at 𝑥(𝑝). Consequently, any directional derivative

𝜕ℎ (𝑓 /𝑥)(𝑥(𝑝)) vanishes. Now, let 𝑣 any tangent vector at 𝑝, the real value 𝑑𝑝 𝑓 (𝑣) equals 𝑣(𝑓 ) and this last value equals a directional derivative of the representation 𝑓 /𝑥 at 𝑥(𝑝) so 𝑑𝑝 𝑓 (𝑣) should vanishes. We can infer that the differential 𝑑𝑝 𝑓 should equal the null functional on the tangent space 𝑀𝑝 . As we claimed.

11.7 Differential of a vector function In this section, we generalize the concept of differential for vector valued functions defined on differentiable manifolds.

Definition (differential of a vector function).

We define the

𝑝 of a smooth vector function 𝑓 : 𝑉 → R𝑛 , locally defined 𝑝, as the following linear operator (𝑛-covector)

differential at on

𝑀

at

𝑑𝑝 𝑓 : 𝑇𝑝 𝑀 → R𝑛 : 𝑣 ↦→ 𝑣(𝑓 ) := (𝑣(𝑓 𝑗 ))𝑛𝑗=1 = (𝑑𝑝 𝑓 𝑗 (𝑣))𝑛𝑗=1 . 𝑓 ⋃︁

And we define the differential of

𝑑𝑓 : 𝑉 → 𝑇 *𝑛 𝑀 :=

as the

𝑛-covector

field

Hom(𝑀𝑝 , R𝑛 ) : 𝑞 ↦→ 𝑑𝑞 𝑓,

𝑝∈𝑀

101

11.8.

DERIVATIVE VERSUS DIFFERENTIAL BIS

sending each point

𝑞

of the domain

𝑉

to the operator

𝑑𝑞 𝑓 .

We can associate, with the differential of 𝑓 , the mapping

𝜕𝑓 : 𝑉 → (𝑇 * 𝑀 )𝑛 sending each point 𝑞 of the domain 𝑉 to the ordered family

𝜕𝑞 𝑓 := (𝑑𝑞 𝑓 𝑗 )𝑛𝑗=1 .

Definition (exact vector differential form).

A

𝑛-covector field

𝜔 : 𝑉 → 𝑇 *𝑛 𝑀 is named exact if

𝜔 = 𝑑𝑓 for some smooth function

𝑓 : 𝑉 → R𝑛 .

11.8 Derivative versus differential bis Let us compare the two notions of derivative and differential.

Proposition. every point

𝑝

tangent space

of the

𝑀𝑝 ,

𝑓 : 𝑀 → R𝑛 is a smooth function, then, manifold 𝑀 and every tangent vector 𝑣 of

If

we obtain

𝑓𝑝′ (𝑣) =

𝑛 ∑︁ (𝑑𝑓 𝑗 )𝑝 (𝑣)(𝜕/𝜕𝑟)𝑗 |𝑓 (𝑝) , 𝑗=1

102

for the

CHAPTER 11.

where

𝑟


represents the identity chart of the real Euclidean

Proof. Since the derivative 𝑇𝑓 (𝑝) R𝑛 , then there exists a real

𝑓𝑝′ (𝑣)

=

𝑛-space.

𝑓𝑝′ (𝑣) belongs to the tangent space 𝑛-vector 𝑎 such that

𝑛 ∑︁

𝑎𝑗 (𝜕/𝜕𝑟)𝑗 |𝑓 (𝑝) .

𝑗=1

To evaluate the coefficient 𝑎𝑘 , apply both sides of the above expression to the projection 𝑟𝑘 , we obtain:

𝑎

𝑘

=

𝑛 ∑︁

𝑎𝑗 (𝜕/𝜕𝑟)𝑗 |𝑓 (𝑝) (𝑟𝑘 ) =

𝑗=1

= 𝑓𝑝′ (𝑣)(𝑟𝑘 ) = = 𝑣(𝑟𝑘 ∘ 𝑓 ) = = 𝑣(𝑓 𝑘 ) = = (𝑑𝑓 𝑘 )𝑝 (𝑣), for every 𝑘 , as we claimed.

103

11.8.

DERIVATIVE VERSUS DIFFERENTIAL BIS

104

Chapter 12 Covector frames 12.1 Change of covector frames Proposition (Transition matrix for covector frames). (𝑈, 𝑥)

and

(𝑉, 𝑦)

Suppose

represent two coordinate charts on a manifold

Then, we see

𝑑𝑥𝑖 |𝑝 =

𝑚 ∑︁

𝑀.

(𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 𝑑𝑦 𝑗 |𝑝 ,

𝑗=1

for every

𝑝

in

𝑈 ∩𝑉.

Consequently, we obtain 𝑖

𝑑𝑥 =

𝑚 ∑︁

(𝜕𝑥𝑖 /𝜕𝑦)𝑗 𝑑𝑦 𝑗 ,

𝑗=1

on the open intersection

𝑈 ∩𝑉 .

In other terms, by using the coordinate

brackets, we can write

[𝑑𝑝 𝑥𝑖 | 𝜕𝑝 𝑦] = (𝜕𝑥𝑖 /𝜕𝑦)|𝑝 , for every

𝑝

in

𝑈 ∩𝑉

and every index

105

𝑖

of

𝑥.

12.1.

CHANGE OF COVECTOR FRAMES

It suffices to prove that the two sides of the equality act in the same way on the same frame of the tangent space. For, let Proof.

𝑋𝑘 := (𝜕/𝜕𝑥)𝑘 |𝑝 , for every index 𝑘 ; we see

𝑑𝑥𝑖 |𝑝 (𝑋𝑘 ) = 𝑋𝑘 (𝑥𝑖 ) = = (𝜕/𝜕𝑥)𝑘 |𝑝 (𝑥𝑖 ) = = 𝛿𝑖𝑘 and 𝑚 ∑︁

𝑚 ∑︁ (𝜕𝑥 /𝜕𝑦)𝑗 |𝑝 𝑑𝑦 |𝑝 (𝑋𝑘 ) = (𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 𝑋𝑘 (𝑦 𝑗 ) = 𝑖

𝑗

𝑗=1

=

𝑗=1 𝑚 ∑︁

(𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 (𝜕/𝜕𝑥)𝑘 |𝑝 (𝑦 𝑗 ) =

𝑗=1

=

𝑚 ∑︁

(𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 (𝜕𝑦 𝑗 /𝜕𝑥)𝑘 |𝑝 =

𝑗=1

= [(𝜕𝑥/𝜕𝑦)|𝑝 (𝜕𝑦/𝜕𝑥)|𝑝 ]𝑖 𝑘 = = 𝛿𝑖𝑘 , as we claimed. It appears extremely interesting to confront the transition rules for tangent vectors and covectors. We can write

[(𝜕/𝜕𝑥)𝑖 |𝑝 | (𝜕/𝜕𝑦)|𝑝 ] = (𝜕𝑦/𝜕𝑥)𝑖 |𝑝 , 106

CHAPTER 12.

COVECTOR FRAMES

and

[𝑑𝑝 𝑥𝑖 | 𝜕𝑝 𝑦] = (𝜕𝑥𝑖 /𝜕𝑦)|𝑝 , for every point 𝑝 in the intersection 𝑈 ∩ 𝑉 and every index 𝑖 of 𝑥. Or, in coordinate fractional notation:

(𝜕/𝜕𝑥)𝑖 |𝑝 = (𝜕𝑦/𝜕𝑥)𝑖 |𝑝 , (𝜕/𝜕𝑦)|𝑝 and

𝑑𝑝 𝑥𝑖 /𝜕𝑝 𝑦 = (𝜕𝑥𝑖 /𝜕𝑦)|𝑝 , for every point 𝑝 in the intersection 𝑈 ∩ 𝑉 and every index 𝑖 of the chart 𝑥. Moreover, for what concerns the change from basis 𝜕𝑝 𝑥 to basis 𝜕𝑝 𝑦 we can write 𝜕𝑝 𝑥/𝜕𝑝 𝑦 = (𝜕𝑥/𝜕𝑦)|𝑝 , for every 𝑝 in 𝑈 ∩ 𝑉 .

12.2 Generalized contravariant vectors To cover completely the variety of tensors considered in the Einstein’s works, we need to introduce a concept of generalized contravariant vector.

Definition (generalized contravariant vectors). smooth space

𝑚-manifold 𝑀

𝑉.

and the

𝑚-Cartesian

power

We shall call a mapping

𝑤 : 𝒜𝑝 → 𝑉 𝑚 : 𝑤(𝑥) = (𝑤(𝑥)𝑖 )𝑚 𝑖=1 , 107

𝑉𝑚

Consider a of a vector

12.2.

a

GENERALIZED CONTRAVARIANT VECTORS

contravariant vector on 𝑀 at 𝑝 with values in 𝑉 𝑚 if we see 𝑚 ∑︁ 𝑤(𝑥) = (𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 𝑤(𝑦)𝑗 , 𝑖

𝑗=1

for every couple of charts

𝑖

𝑥, 𝑦

at

𝑝, on the manifold 𝑀

and every index

of the Cartesian power.

The first example of contravariant vector considered by Albert Einstein in his paper on General Relativity is indeed a generalized contravariant vector with values on the 𝑚 Cartesian power of the cotangent space, the below one.

12.2.1

Einstein’s cotangent bases

Consider a smooth manifold 𝑀 , a point 𝑝 of 𝑀 and a smooth local coordinate system 𝑥 about the point 𝑝. The ordered family of differentials 𝜕𝑝 𝑥 := (𝑑𝑝 𝑥𝑗 )𝑚 𝑗=1 is an element (vector) of the 𝑚-Cartesian power

(𝑀𝑝* )𝑚 of the cotangent space 𝑀𝑝* . It is, indeed, a basis of the cotangent space 𝑀𝑝* . We call the above ordered basis 𝜕𝑝 𝑥 the cotangent basis induced by the chart 𝑥 at the point 𝑝, on the manifold 𝑀 .

We can now state and prove the following Einstein’s theorem. 108

CHAPTER 12.

Theorem.

COVECTOR FRAMES

Consider the mapping

𝑤 : 𝒜𝑝 → (𝑀𝑝* )𝑚 : 𝑤(𝑥) = 𝜕𝑝 𝑥, 𝑥 around 𝑝 to the cotangent basis induced by 𝑥 itself 𝑝. Then 𝑤 reveals a contravariant vector on 𝑀 at 𝑝 with (𝑀𝑝* )𝑚 .

sending each chart at the point values in

Proof.

Indeed, we have

𝑤(𝑥)𝑖 = (𝜕𝑝 𝑥)𝑖 = = 𝑑 𝑝 𝑥𝑖 = 𝑚 ∑︁ = (𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 𝑑𝑝 𝑦 𝑗 = = =

𝑗=1 𝑚 ∑︁ 𝑗=1 𝑚 ∑︁

(𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 (𝜕𝑝 𝑦)𝑗 = (𝜕𝑥𝑖 /𝜕𝑦)𝑗 |𝑝 𝑤(𝑦)𝑗 ,

𝑗=1

for every couple of charts 𝑥, 𝑦 on the manifold 𝑀 .

12.3 Generalized covariant vectors In a perfectly analogous way, we can define the generalized covariant vectors.

Definition (generalized covariant vectors). 𝑚-manifold 𝑀

and the

𝑚-Cartesian

power

𝑉

𝑚

of a vector space

We shall call a mapping

𝑤 : 𝒜𝑝 → 𝑉 𝑚 : 𝑤(𝑥) = (𝑤(𝑥)𝑖 )𝑚 𝑖=1 , 109

Consider a smooth

𝑉.

12.3.

GENERALIZED COVARIANT VECTORS

covariant vector on 𝑀 at 𝑝 with values in 𝑉 𝑚 , or with components in 𝑉 , if we see a

𝑚 ∑︁ 𝑤(𝑥)𝑖 = (𝜕𝑦 𝑗 /𝜕𝑥)𝑖 |𝑝 𝑤(𝑦)𝑗 , 𝑗=1

for every couple of charts

𝑖

of the Cartesian power

𝑥, 𝑦 at 𝑝, on the manifold 𝑀 𝑉 𝑚.

and every index

Also in the present case, we can propose a noble example. If (𝑈, 𝑥) and (𝑉, 𝑦) represent two coordinate charts on a manifold 𝑀 at a point 𝑝, then, we saw that

(𝜕/𝜕𝑥)𝑖 |𝑝 =

𝑚 ∑︁

(𝜕𝑦 𝑗 /𝜕𝑥)𝑖 |𝑝 (𝜕/𝜕𝑦)𝑗 |𝑝 ,

𝑗=1

for every index 𝑖 of 𝑥. Therefore, we can deduce the following theorem.

Theorem.

Consider the mapping

𝑤 : 𝒜𝑝 → (𝑀𝑝 )𝑚 : 𝑤(𝑥) = 𝜕/𝜕𝑥|𝑝 , 𝑝 of a manifold 𝑀 𝑝. Then, 𝑤 reveals a generalized covariant vector on 𝑀 at the point 𝑝 with values in the Cartesian power (𝑀𝑝 )𝑚 (i.e, with components in the tangent space 𝑀𝑝 ). sending each local coordinate chart about a point into its induced tangent frame at

110

CHAPTER 12.

Proof.

COVECTOR FRAMES

Indeed, we have

𝑤(𝑥)𝑖 = (𝜕/𝜕𝑥)𝑖 |𝑝 = 𝑚 ∑︁ = (𝜕𝑦 𝑗 /𝜕𝑥)𝑖 |𝑝 (𝜕/𝜕𝑦)𝑗 |𝑝 = =

𝑗=1 𝑚 ∑︁

(𝜕𝑦 𝑗 /𝜕𝑥)𝑖 |𝑝 𝑤(𝑦)𝑗 ,

𝑗=1

for every couple of charts 𝑥, 𝑦 on the manifold 𝑀 and every index 𝑖 of the smooth chart 𝑥.

111

12.3.

GENERALIZED COVARIANT VECTORS

112

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