Differential Geometry and Relativity Theories: tangent

Volume II, Issue1(2), Summer 2016

DOI: http://dx.doi.org/10.14505/jmef.v2.1(2).05

Dierential Geometry and Relativity Theories: tangent vectors, derivatives, paths, 1-forms David Carfì Department of Mathematics, University of California Riverside, USA [email protected]

Abstract.

In this lecture note, we focus on some aspects of smooth manifolds, which appear

of fundamental importance for the developments of dierential geometry and its applications to Theoretical Physics, Special and General Relativity, Economics and Finance. In particular we touch basic topics, for instance:

1.

denition of tangent vectors;

2.

change of coordinate system in the denition 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;

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.

dieomorphisms 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;

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Journal of Mathematical Economics and Finance

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.

dierential

28.

dierential of a function;

29.

derivative versus dierentials;

30.

critical points of real functions;

31.

dierential of a vector function;

32.

change of covector frames;

33.

generalized contravariant vectors;

34.

generalized covariant vectors.

1-forms;

Keywords:

smooth manifolds; tangent vectors; coordinate systems; tangent frames; contravariant vectors; covariant vectors; invariant scalar; tangent applications, local Jacobian; paths on manifolds.

0. Preliminaries: charts, atlases, manifolds 0.1 Charts and atlases We begin with the concept of local chart.

Denition (of local chart). x

on

S

Consider a Hausdor topological space

is a topological isomorphism of an open subset

open subset

x(U )

of some Euclidean space

U

S.

A local chart

of the topological space

S

onto an

Ex .

Now, we can dene the fundamental object of atlas.

C k (k

Denition (of atlas).

Consider a Hausdor topological space

positive integer or innity) on the topological space

S

S.

is a collection

An atlas of class

A

of charts on

satisfying the following conditions:

•

the collection of all chart domains of the atlas

•

for any charts

x, y

of the atlas

A,

A

with domain

covers the topological space

U, V

S;

respectively, the map

y/x : x(U ∩ V ) → y(U ∩ V ) dened by

x0 7→ (y ◦ x− )(x0 ) is a

C k -isomorphism (C k -dieomorphism,

86

if

k

is dierent from 0 or innity).

S


0.2 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 dened by

x(p) 7→ 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.

0.3 Manifolds In the conditions of the preceding denition of atlas the pair (S, A) is called a manifold of class C k , if no other atlas contains properly A. 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 denition 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 non-empty 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 xed space E .

0.4 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 arm that the chart y is compatible with the atlas A if each transition map y/x (dened on the convenient intersection as in the atlas denition) is a C k -isomorphism. 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.

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An equivalence class of atlases of class manifold on

Ck

on

S

is said to dene a structure of

Ck-

S.

A k -dierentiable structure on a topological space denes a maximal atlas on that space.

0.5 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 xed m, then we say that the manifold M is m-dimensional.

0.6 Coordinate systems In m-dimensional manifolds, all charts

x : U → x(U ) are dened 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.

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

0.8 Induced dierentiable 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 .

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

f : M → Rn 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 dened 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 denition 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−

(a map from the open subset x(f − (V ) ∩ U ) of Rm to 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 . This is equivalent to say that a continuous map

f :M →N is smooth, at a point p of M if there exists a chart (V, y) about f (p) in 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.

1. Tangent vectors 1.1 Denition of tangent vectors Here, we introduce the fundamental concept of a tangent vector on dierentiable (smooth) manifolds.

Denition (of tangent vector). m,

Consider a smooth real manifold

M , with dimension

and the collection

Cp∞ (M, R), of all smooth real functions locally dened upon open neighborhoods of a point dene a tangent vector on the manifold

M

at a point

p

of

M

p

in

M.

We

as a real functional

v : Cp∞ (M, R) → R such that, for each local coordinate system belonging to the real Euclidean

m-space,

x

of the manifold, we can nd a vector

such that the value of the functional

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v,

v(x),

at any locally


f at p, x-representation

dened smooth function

v(x),

of the

equals the directional derivative, with respect to the vector

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,

the

is dened a tangent vector at the point p, if for every chart x in Ap , there exists m ∞ a vector vx of the Euclidean space R such that, for any function f of Cp (M ), we see: functional

v

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

v(f ) = dx(p) (f /x)(v(x)) = (f /x)0 (x(p))(v(x)), in terms of dierentiation.

1.2 Basic properties Theorem.

The value of a tangent vector

depends only on the germ of the function on a neighborhood of the point

p

f,

v

at a point

p,

upon a smooth function

that is, if two smooth functions

f

and

g

f

coincide

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

constant around the point

p

p

on a test function at

p,

which remains

is 0.

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

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

1

f, g

of smooth functions.

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-dened by f/x := f ◦ x− : x(U ∩ V ) → R : x0 7→ f (x− (x0 )).

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

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Proof.

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

Theorem.

Any tangent vector

v

at a point

p,

on a smooth manifold

M,

denes a

linear functional (compact support distribution)

u : C ∞ (M, R) → R : f 7→ v(f ), restriction of the tangent vector

v

to the vector space

derivative in any local coordinate chart at

p.

C ∞ (M, R),

acting as a directional

Vice versa, any linear functional

u : C ∞ (M, R) → R, C ∞ (M, R), acting as a tangent vector v such that

on the vector space at

p,

denes a

directional derivative in any local coordinate chart

v(f ) = u(g), for any smooth locally dened function the same germ of

f

at

p

(i.e.

g

f

equal to

at

f

p

and any smooth globally dened function

on a convenient neighborhood of

g

with

p.

Proof. Indeed, let f be a smooth real function dened 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 nd a smooth function g dened upon M such that: ¯ is contained in V ; • it equals zero out of an open neighborhood U whose closure U • 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 dene the value of u on f as

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

v(f ) := u(f˜g), for every f locally dened at p. Clearly, the denition is well posed because it does not depend on the arbitrate choice of the extension f˜ and the test function g , but only upon the germ of f at p.

1.3 Example: tangent vectors upon R Consider a real number p and the point derivative functional Dp , on the real line, sending each smooth real function - locally dened on the real line around p - to its derivative f 0 (p) at the point p, that is Dp (f ) = f 0 (p), for every dierentiable function f . For every coordinate system x around p, we easily obtain that Dp (f ) = ∂x0 (p) (f /x)(x(p)),

91


for every dierentiable function f . Indeed,

∂x0 (p) (f /x)(x(p))

= x0 (p)∂1 (f /x)(x(p)) = = x0 (p)(f /x)0 (x(p)) = = x0 (p)(f 0 ◦ x− )(x(p))(x− )0 (x(p)) = = x0 (p)f 0 (x− (x(p))x0 (x− (x(p)))−1 = = f 0 (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 ) = ∂ax0 (p) (f/x )(x(p)), for every smooth function f . An alternative, shorter (but more sophisticated) proof follows:

∂x0 (p) (f /x)(x(p))

= x0 (p)∂1 (f /x)(x(p)) = =

x0 (p)(∂f /∂x)p =

= =

x0 (p)(df /dx)(p) = x0 (p)(f 0 (p)/x0 (p)) =

=

f 0 (p),

where we have used:

• the denition of the derivative

df /dg := f 0 /g 0 ,

of a function f with respect to another function g ;

• the property that the derivative ∂f /∂x|p := (f/x )0 (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).

1.4 Change of coordinate system We shall here that, in order to verify the tangency property of a functional, we need not to verify the tangency property itself for any coordinate system but it just suces to see it for only one of them. Note the following basic property.

Property.

If a functional

v : Cp∞ (M, R) → R acts like a directional derivative in a coordinate system at a point tional derivative in all coordinate systems at

p.

92

p,

then it acts like a direc-


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.

1.5 Action of tangent vectors on coordinate systems In a certain sense, we have already extended the domain of action of the tangent functional v - by adding, to the collection of locally dened smooth real functions at p, the collection of all local coordinate systems about p - when we decided to dene 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 dened real functions. Let us note that the i-th component xi of a coordinate system x is a locally dened 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 represents the i-projection of the Euclidean m-space and ei|x represents its restriction from x(U ) to x(U ). Therefore, we obtain m m v(x) = (v(x)i )i=1 = (v(xi ))i=1 .

Therefore, we can state the following proposition.

Proposition. m-tuple

The action of the tangent vector

of the single actions of the tangent vector

v

v

on a coordinate system x equals the xi of the coordinate

on the components

system itself.

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

f/x := f ◦ x− of the real function f , at the point x(p), that is:

v(f ) = ∂v(x) (f /x)(x(p)).

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Denition (representative of a tangent vector). denition, we call representative of

v,

In the conditions of the preceding

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 dened on M .

Denition (action of a tangent vector v on a vector function). a tangent vector

v

at

p,

of a manifold

M,

The action of

on a vector function,

f : V → W, p, where V and W represent two open subsets of the manifold and of n-space, equals the n-tuple of the single actions of the tangent vector v on j components f of the vector function f locally dened at

the

Euclidean

the

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

1.6 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, dened by

(v + w)(f ) = v(f ) + w(f ) and

(av)(f ) = av(f ), for every couple of tangent vectors v, w and every smooth locally dened function f . Now, we state and prove a theorem regarding, more specically, the algebraic structure of tangent spaces of smooth manifolds. At this aim, we introduce here the fundamental concept of dierential dp x of a chart x at a point p of its domain, dened as the below linear operator associating with any tangent vector at the point p its representation in the xed chart x, dp x : Mp → Rm : v 7→ vx . It appears as a simple and straightforward generalization of the dierential dp f of a smooth real function f at a point p of its domain, dened as the below linear functional associating with any tangent vector v at the point p its value v(f ) upon the xed smooth function f ,

dp f : Mp → R : v 7→ v(f ).

Theorem (structure of tangent spaces). manifold

M,

Fixed a coordinate chart

x

of a smooth

the application

dp x : Mp → Rm : v 7→ vx , reveals a linear isomorphism, so that, the algebraic dimension of tangent space at any point of a smooth

m-manifold

equals

m.

Moreover, the inverse

(dp x)− : Rm → Mp

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of the operator

dp x

is dened by

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

h

belonging to

Rm

and every smooth function

f

locally dened at

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 dened by

v(f ) := ∂v∗ (f/x )(x(p)), satisfy the following equality

(dp x)(v) = v∗ , as we claimed.

1.7 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 dene 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 dene the action of such pointed vector on usual dierentiable 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 dierentiable 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 dene a natural isomorphism x0p , depending on a chosen local chart x, of a the tangent space Mp onto the tangent space Tx(p) R, by x0p : v 7→ (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 dierentiable manifold, the point partial derivative ∂i |p - sending each function f , locally dened 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

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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 ij = ei (ej ), for each index pair i, j .

Remark.

In the denition 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, nally, 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. 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.

1.8 Tangent vectors determined by local 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 .2 Then, for every index i of the coordinate system x, we can dene a tangent vector at p ∂ ∂ ∂ : Cp∞ (M, R) → R |p = = ∂x ∂x p,i ∂x i p i 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 for every smooth real function f locally dened on M about the point p. Here we recall that 2

A local coordinate system on M centered at p is a function dened 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 7→ x(q) = (xi (q))m i=1 .

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

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• 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, dened 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 denition. The above denition depends upon the entire coordinate system x and on the index i previously selected. Clearly, the just dened functional

v = ∂/∂xi |p appears a bona de 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 denition.

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

the

i-th

∂ ∂x

(p) : Cp∞ (M, R) → R : f 7→ ∂i (f /x)(x(p)), i

tangent vector determined by the coordinate system

x

at the point

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

1.9 Tangent frames determined by local charts On the other hand, 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 .

97

p.


Denition (of 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 justied by the following theorem.

Theorem. vector

v

Let

x

be a chart of a manifold

in the tangent space

Mp

M

around a point

p.

Then, any tangent

can be uniquely written as a linear combination

v=

X

v(x)(∂/∂x)p ,

of the family

(∂/∂x)p , where the coecient system

v(x)

remains dened 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 dinate system of the tangent vector

v

v

in the coordinate system

in the basis

(∂/∂x)p ,

x,

becomes the coor-

that is

v(x) = [v | (∂/∂x)p ], where by

[v|b]

we denote the coordinate system of a vector

v

in a basis

b.

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)) = X = v(x)∂(f /x)(x(p)) = X = v(x)(∂/∂x)p (f ),

for every f . Concluding we can write:

v=

X

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.

1.10 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:

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• 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;

• the (tangent) frame eld induced by the chart x upon its domain U , ∂/∂x : U → TU (M )m : p 7→ ∂/∂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 eld (∂/∂x)i : U → TU (M ) : p 7→ (∂/∂x)i |p , for every index i of the chart x.

Proposition (Transition matrix for local frames). represent two coordinate charts on a manifold

(∂/∂x)i |p =

M.

Suppose

(U, x)

and

(V, y)

Then, we see

m X (∂y j /∂x)i |p (∂/∂y)j |p , j=1

for every

p

in

U ∩V.

Consequently, we obtain

(∂/∂x)i =

m X (∂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 , for every

p

in

U ∩V

and every index

i

of

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 eld a : U ∩ V → Rm,m dened by

a(p) = [aji (p)]m j,i=1

such that

(∂/∂x)i =

m X

aki (∂/∂y)k

k=1

on U ∩ V , that is,

(∂/∂x)i |p =

m X

aki (p)(∂/∂y)k |p

k=1

99


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

(∂y j /∂x)i

m X

=

aki (∂y j /∂y)k =

k=1 m X

=

aki δ jk =

k=1 aj i ,

= on U ∩ V , as we claimed.

2. Contravariant and covariant vectors 2.1 Tangent vectors and contravariant vectors In the preceding sections, we have already associated a contravariant vector v¯ with any tangent vector v , namely, the function sending any local coordinate system x at p to the representation v(x) of the tangent vector v in the coordinate system x. Let us recall the denition of contravariant vector.

Denition (of contravariant vector). M

with atlas

A,

a point

p

of the manifold

M

Consider an

m-dimensional smooth manifold

and the coordinate system collection

Ap = {y ∈ A : p ∈ domy}, of all the charts of

v,

M

at

p.

Then, a contravariant vector upon

M

at the point

p,

say

is a mapping

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

x

and

y

of

Ap ,

we obtain

v(y) = dx(p) (y/x)(v(x)), where

write:

y/x

denotes the transition mapping from

x

to

y.

In terms of the matrix associated with the dierential of the transition map, we can

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). So, we can obtain

v i (y)

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

=

j=1 m X

(∂y i /∂x)j |p v j (x)

j=1

=

(∂y i /∂x)|p · v(x),

100


which provides the contravariance denition in terms of components.

2.2 Covariant vectors Let us recall the denition of covariant vector.

Denition (of covariant vector). with atlas

A,

a point

p

of the manifold

M

Consider an

m-dimensional

smooth manifold

M

and the coordinate system collection

Ap = {y ∈ A : p ∈ domy}, 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

x/y

denotes the transition mapping from

y

to

x

and where

dy(p) (x/y)∗ represents the adjoint operator of the dierential

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 m-dierential manifold M with atlas A. For every index i, we dene the i-th component of the covariant vector w, as the application wi : Ap → R : x 7→ w(x)i , whose variation (action) is determined by

wi (y)

=

m X

(∂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 dierential

dy(p) (x/y), with respect to the standard scalar product in Rm , dened by dy(p) (x/y)∗ (k) | h = k | dy(p) (x/y)(h) ,

101


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.

2.3 The gradient of a real function Consider the vector function w dened on local charts centered at a point p of an m-dierential manifold M with atlas A,

w : Ap → Rm : x 7→ (∂f /∂x)p , associating with any coordinate system x the gradient of a certain real function

f : V → R, locally dened at p, with respect to x. This function w reveals a covariant vector. Indeed, for any charts x, y about p, we know

(∂f /∂x)i |p

=

m X

(∂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.

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

(v|w) : Ap → R : x 7→ (v(x)|w(x)). We shall see that this scalar product reveals a scalar invariant at p, that is a constant mapping from Ap to R. Our proof essentially reproduce a classic Einstein's proof from the rst paper about General Relativity.

Theorem. v

at

p

Consider a dierential manifold

w

and a covariant vector

M,

a point

p

of

M,

a contravariant vector

at the same point. Then, the application

(v|w) : Ap → R : x 7→ (v(x)|w(x)), Euclidean scalar product of the two above vectors, reveals a scalar invariant at constant mapping from

Ap

to

R.

102

p,

that is a


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

=

m X

For, we observe that

v i (y)wi (y) =

i=1

=

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

=

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

=

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

i=1

i=1 j=1

=

=

=

k=1

k=1 j=1 i=1 m X m X

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

k=1 j=1 m X m X

δ kj v j (x)wk (x) =

k=1 j=1 m X 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 , of the Jacobian matrices associated with the two mutually inverse transition maps

y/x,

is the identity

(m, m)

matrix

x/y

and

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)),

103


by proting 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)),

by proting of the behavior of transpose matrices in scalar products. So, we conclude the proof.

3. Tangent mappings 3.1 Tangent applications Consider a dierentiable function

f : Rm → Rn between Euclidean spaces and a point p of its domain Rm . In nite dimensional vector calculus, the Jacobian matrix of f at the point p, dened by Jf (p) := (∂i f j (p))n,m j,i=1 , reveals the (n, m)-matrix associated with the derivative f 0 (p) of the function f at the point p. The derivative of the function f at the point p is the linear operator dened as follows:

fp0 : Rm → Rn : v 7→ ∂v (f )(p), where ∂v (f )(p) denotes the usual directional derivative, with respect to v , of the function f , at the point p. a map

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

f :M →N between manifolds M and N , to look again for a linear transformation. Previously, we dened a vector space at each point of a manifold, locally representing the manifold itself, this suggests to dene a linear transformation

fp0 = 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 fp0 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 identied with the vector space of all pointed vectors (q, v), with v in Rk . In other terms, we require that

fp0 (p, v) = (f (p), Jf (p)v)

104


for all vector v in Rm . Now, if

g : Rn → R

is dierentiable, then by the Chain Rule,

fp0 (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 ).

= This motivates the following denition.

Denition (of derivative). m

and

n

Let

M p

respectively. Consider a point

N M,

and

denote two smooth manifolds of dimensions

of

an open neighborhood

U

of

p

in

M

and a

map

f :U →N 3 dierentiable at the point p. Then, we dene the derivative of

f

at

p

as the map

fp0 : Mp → Nf (p) given by

fp0 (v)(g) := v(g ◦ f ), where

g

lies in the vector bered space

Cf∞(p) (N ), of smooth functions locally dened on

N

at the point

f (p),

and

v

lies in the tangent space

Mp .

It is clear from the denition that fp0 is a linear transformation. We shall avoid to denote the derivative fp0 by the symbol f 0 (p). We shall reserve the latter symbol for the classic Frechet derivative, when it applies.

3.2 Local Jacobian matrices In this section we examine the local representation of derivatives via Jacobian matrices. 3

A map between two dierentiable manifolds f :U →N

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

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

105


Proposition. and

∂/∂y|f (p)

p

With notation as in the above Denition, let

x

be a coordinate map

a coordinate map around the point f (p) in the 0 manifold N . Then, the matrix of the derivative fp , with respect to the induced bases ∂/∂x|p around the point

of the open part

U, y

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 fp0 , obtaining a family of tangent vectors at f (p), and then, we need to apply the family fp0 (X) to the coordinate function y j to get the j -th row of the desired associated matrix. In other terms, the matrix of the derivative fp0 , with respect to the bases X = ∂/∂x|p and Y = ∂/∂y|f (p) is the matrix

[fp0 ]YX = [fp0 (X)|Y ], whose j -th row is the vector

[fp0 (X)|Y ]j = ([fp0 (Xi )|Y ]j )m i=1 , or, equivalently, whose i-th column is the system of coordinates

[fp0 (Xi )|Y ] = fp0 (Xi )(y). Therefore, the (j, i) element of the matrix [fp0 ]YX is

([fp0 ]YX )ji

=

[fp0 (Xi )|Y ]j =

=

fp0 (Xi )(y)j =

=

fp0 (Xi )(y j ).

Moreover, we immediately see that

fp0 (∂/∂xi |p )(y j )

=

(∂/∂x)i |p (y j ◦ f ) =

= ∂i (y j ◦ f ◦ x− )(x(p)) = =

(Jx(p) (y ◦ f ◦ x− ))ji ,

for every index j and i. Concluding, we can write

[fp0 ]YX = Jx(p) (y f/x ), as we claimed.

106


3.3 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 denition 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 dierential of the

immersion

j : U → M : u 7→ 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

jp0 (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 dened at the point p. We used the property arming 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.

3.4 Chain rule Now we can prove the most important property of the tangency: the chain rule.

Theorem (chain rule).

Let consider two dierentiable mappings

f, g

between mani-

folds:

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

g◦f is dierentiable, and

(g ◦ f )0p = gf0 (p) ◦ fp0 , or, equivalently

(g ◦ f )∗p = g∗f (p) ◦ f∗p .

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 )0p (w)(h) = w(h ◦ (g ◦ f )) and

(gf0 (p) ◦ fp0 )(w)(h)

=

gf0 (p) (fp0 (w))(h) =

=

(fp0 (w))(h ◦ g) =

=

w(h ◦ g ◦ f ),

107


as we claimed.

3.5 Dieomorphisms and derivatives One particular case of the Chain Rule, at once, oers us the following property about dieomorphisms. A dieomorphism between two manifolds is an invertible mapping, between the two manifolds, which is smooth together with its inverse.

Theorem.

Consider a dieomorphism

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

p

of the domain manifold

M,

the derivative

fp0 : 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,

(fp0 )− = (f − )0f (p) .

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

g ◦ f = IM and

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

ITp M = (IM )0p = (g ◦ f )0p = gf0 (p) ◦ fp0 , and, setting q := f (p) (so that p = g(q)), we obtain 0 fp0 ◦ gf0 (p) = fg(q) ◦ gq0 = (f ◦ g)0q = (IN )0q = ITq N .

Hence, the derivatives fp0 and gq0 are vector space isomorphisms. On the other hand, the dieomorphism g equals the inverse of f ,

g = f −, and, from we infer

fp0 ◦ gq0 = ITq N , (fp0 )− = (g)0q = (f − )0f (p) ,

as we claimed. .

108


Corollary (Invariance of dimension under dieomorphism). U

of the real Euclidean n-space reveals dieomorphic to an open set m space R , then dimension n equals dimension m.

V

If an open subset

of the Euclidean

m-

Proof. Let f : U → V be a dieomorphism of the manifold U to the manifold V and let p be a point of U . By the above corollary, the derivative fp0 : Up → Vf (p) , of the dieomorphism 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.

3.6 Transformation of 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 dieomorphism onto its image, then the dierential

x0p : Mp → x(U )q , 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.

Let

(U, x)

be a chart about a point

p

in a manifold

M.

Then

x0p (∂/∂xi |p ) = ∂i |x(p) , for every index

Proof.

i

of

x.

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

x0p (∂/∂xi |p )(f ) = (∂/∂xi |p )(f ◦ x) = ∂i |q (f ◦ x ◦ x− ) = ∂i |x(p) (f ), as we claimed. .

4. Paths on manifolds 4.1 Paths on a Manifold Denition (smooth path).

A smooth path in a manifold

smooth map

p : I → M,

109

M

is by denition a


from some open interval of the real line the parameter space of the path

p

I

into the manifold

and each element of

M . The interval I will be called I will be named 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 the point p0 at the parameter 0, which means

point

p0

of

M if the path p passes through

p(0) = p0 .

Denition (vector derivative of a path). M,

Consider a smooth path in a manifold

a smooth map

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

I.

I

into the manifold M . Let y represent the identity p0 (y0 ) of the path p at the point y0 ∈ I

The vector derivative

is dened as the tangent vector

p0 (y0 ) := p∗y0 of the tangent space

d   dy y0

Tp(y0 ) M .

In the conditions of the above denition, the tangent vector d 0 , p (y0 ) := p∗y0 dy y0 acts as it follows

p0 (y0 ) : f 7→ (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 . obtain

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

p0 (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 p0 (y0 ) is the vector derivative of the path p at the point p(y0 ).

110


Alternative notations for the vector derivative p0 (y0 ) are

dp (y0 ) dy and

d   p, dy y0

where y represents the identity function on the interval I .

4.2 Vector derivative of a path by a re-parametrization Denition (vector derivative of a path with respect to a re-parametrization). Consider a smooth path in a manifold

M,

a smooth map

p: Y →M Let

z

represent a local

with respect to the coordinate system

z

is dened as the

from some open interval of the real line coordinate system of the interval

Y.

Y

into the manifold

M.

The vector derivative

(dp/dz)(y0 ) of the path

p

at the point

y0 ∈ I ,

tangent vector

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

d   dz y0

Tp(y0 ) M .

In the conditions of the above denition, the tangent vector  d (dp/dz)(y0 ) := p∗y0 ,  dz y0 acts as it follows

x 7→ 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 ).

4.3 Tangent derivative versus calculus derivative When

p : I → Rn

represents a smooth path in the Euclidean space Rn , the notation p0 (y0 ) for the vector derivative could generate some confusion, because, in Calculus, p0 (y0 ) represents, instead, the nvector whose components are the slopes of the graphs of the single components pj at the point y0 , which are real numbers. Therefore, in Calculus, p0 (y0 ) is an n-vector and not a derivation

111


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 denition, p0 (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 p0 (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 p0 (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).

identity function of the real line and let

Let x represent the

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

p0 (y0 ) = p(y ˙ 0 ) (d/dx)|p(y0 ) , for every y0 in I .

4.4 An elementary example Let m, n represent two natural numbers. Dene the path

c : R → R2 by

c(p) = (pm , pn ), for every real number p. Then, the point derivation c0 (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

c0 (p) = a∂1 |q + b∂2 |q . To compute the coecient a, we evaluate the above linear combination on the rst canonical projection x := e1 , obtaining

a

=

(a∂1 |q + b∂2 |q )x =

= c0 (p)(x) = = c∗p (d/dz|p )(x) = = d/dz|p (x ◦ c) = = d/dz|p (z m ) = = mpm−1 .

112


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

b =

Thus,

(a∂1 |q + b∂2 |q )y =

=

c0 (p)(y) =

=

c∗p (d/dz|p )(y) =

=

d/dz|p (y ◦ c) =

=

d/dz|p (z n ) =

=

npn−1 .

c0 (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 [c0 (p)|∂(q)] = . npn−1

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

Proposition (Vector derivative of a path in local coordinates).

Let

p:I→M be a smooth path in a smooth a point

p0 = p(q0 )

of

M,

with

m-manifold M , and let (U, x) q0 in the interval I . Set

be a local coordinate chart about

pix := xi ◦ p, for the i-th component of the path

p

in the chart

by 0

p (q0 ) =

m X

p˙ix (q0 )

i=1

is represented by the

of the Euclidean space Proof.

Then, the vector derivative

∂ ∂x

   i

∂/∂x|p0 of the tangent m-vector 1 p˙x (q0 ), ..., p˙m x (q0 ) ,

Thus, relative to the tangent basis

p0 (q0 )

x.

Rm .

Straightforward.

113

p0 (q0 )

is given

.

p(q0 )

space

Tp0 M ,

the vector derivative


4.6 Paths with a given initial tangent vector In a manifold M , every smooth path p, passing through a point p0 , gives rise to a tangent vector p0 (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). any point

p0

m-manifold M and any tangent vector v interval I , of the real line, centered at 0 and a

of a smooth

there exist an open


For

M p0 ,

smooth path

p:I→M such that

p(0) = p0

and

p0 (0) = v.

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 m X vxi (∂/∂x)i |p0 , v= i=1

at the point p0 , for the convenient uniquely determined coecient 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 dened by xi = ri ◦ x, for every index i of the chart x. To nd a smooth path p passing through p0 and with p0 (0) = v , we use a path q : R → Rm , in Rm , with q(0) = 0 and

q 0 (0) =

m X

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 dened 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 suciently small so that the point q(z0 ) lies in the neighborhood x(U ) of the origin 0, for every z0 in I . Dene the required path p as the below composition

p := x− ◦ p : I → U. Then, we obtain

p(0) = x− (q(0)) = x− (0) = p0 ,

114


and, by the chain rule,

p0 (0)

= = = = = = = =

(x− ◦ q)0 (0) =  d (x− ◦ q)00  = dy 0  d ((x− )0p0 ◦ q00 )  = dy 0  d (x− )0p0 q00 =  dy 0  ! m X ∂  − 0 i (x )p0 vx =  ∂r i 0 i=1  m X ∂  vxi (x− )0p0  = ∂r i 0 i=1 m X ∂ vxi = ∂x i p0 i=1 v,

as we claimed.

4.7 Tangent vectors as vector derivatives of paths We dened a tangent vector at a point p of a manifold as a point directional derivative functional at p. Using paths, we can now interpret a tangent vector geometrically as a vector derivative of a smooth path.

Proposition. any real function

f

Suppose

v

is a tangent vector at a point Cp∞0 (M ). If

p0

of a manifold

M

and consider

in the bered space

p:I→M is a smooth path originating at

p0

with

p0 (0) = v, then

v(f ) = where

y

represents the identity chart on

Proof.

d   (f ◦ p), dy 0

I.

By the denitions of the tangent derivative p0 (0) and recalling that the derivative

p∗ : y0 7→ p0y0 acts as below

p0y0 (w)(f ) = w(f ◦ p),

115


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

v(f )

= p0 (0)(f ) = d = p00 (f ) = dy 0 d = (f ◦ p), dy 0

for every smooth function f locally dened about the point p(0) of M , as we claimed.

4.8 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 fp0 : Np → Mf (p) and dened as follows: if the tangent vector v lives in the tangent space Np , then the derivative fp0 (v) is the tangent vector in the space Mf (p) described by

fp0 (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 denition) and in terms of local coordinates (by the local Jacobian matrix). The next proposition gives another way of computing the derivative fp0 , using paths.

Proposition. N

the manifold

Consider a smooth map

and a tangent vector

v

f :N →M

represent a smooth path originating at the point

v

at the point

p

and let

y

Tp N .

in the manifold

Let the map

N,

with vector derivative

represent the identity map of the open interval

fp0 (v) =

parameter point 0 (note that

I.

Then

d   (f ◦ q). dy 0

fp0 (v) is the f (q(0)) = f (p)).

In other words, the tangent vector

Proof.

p

vector derivative of the image path

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

fp0 (v)

=

p of q :I →N

between manifolds, a point


fp0 (q 0 (0)) =

fp0 (q00 (d/dy|0 )) =  d = (fp0 ◦ q00 )  = dy 0  d = (f ◦ q)00  = dy 0 d  =  (f ◦ q), dy 0 =

116

f ◦q

at the


as we claimed.

5. Covectors and dierentials 5.1 Cotangent vectors In order to introduce the fundamental concept of a dierential 1-form, we need to dene the cotangent space to a point p of a smooth manifold.

Denition (cotangent space).

M be a smooth manifold and p a point of M . M at the point p, denoted also by Tp∗ (M ) or Mp∗ of the tangent space Mp of M at the point p,

Let

The cotangent space of the manifold Tp∗ M , is dened as the algebraic dual space

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

the manifold at the point p. Denition (covector). ector at

An element of the cotangent space

cotangent vectors of

Tp∗ M

is called also a cov-

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.

Denition (cotangent bundle space). collection of all covectors on a manifold

M

T ∗ (M ), we represent the manifold M ), i.e., the below

By the symbol

(at any point

p

of the

union

T ∗ (M ) :=

[

Tp∗ (M ).

p∈M

We call that collection by the name of cotangent bundle space.

The cotangent bundle space is a bered space with respect to the projection

π : T ∗ (M ) → M, dened by

π(ω) = p if ω ∈ Mp∗ ,

for every covector ω .

5.2 Dierential 1-forms Now we can dene the fundamental concept of dierential form on a manifold.

117


Denition (covector eld). smooth manifold

M,

A covector eld, or dierential 1-form, or 1-form, on a

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 elds appear as dual objects of vector elds 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 dene a pairing (scalar product)

(.|.) : (ω, v) 7→ (ω|v) := ω(v), where the action ω(v), of a 1-form on a vector eld v , is point-wise dened as the function

ω(v) : p 7→ ωp (vp ), from M to R. A vector eld v is dened smooth if the real function

v(f ) : p 7→ vp (f ), reveals smooth, for every smooth function f globally dened on M . A dierential form is dened smooth if the function ω(v) reveals smooth for every smooth vector eld v .

Note. We know many reasons motivating the introduction of dierential forms in Differential Manifold Theory: rst of all, dierential forms represents many classes of physical quantities, in Newtonian, Lagrangian and Hamiltonian Mechanics, in thermodynamics, uiddynamics and Einstein Relativity. Moreover, dierential forms stay at the foundations of the Integration Theory in dierential manifolds. Finally, we shall observe that we can pull back dierential forms under smooth maps, in contrast to vector elds, as, in general, we cannot push forward a vector eld under a smooth map. 5.3 Dierential of a function Covector elds appear naturally from the study of local coordinate systems. For, consider an open part V of a manifold M and let

v : V → TM be a smooth vector eld 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 X

vpi (x)(∂/∂x)i |p ,

i=1

118


where the coecient vpi (x) represents the i-th component of the representation vp (x) of the tangent vector vp in the local chart x. The above coecient 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 7→ 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 eld on U Lix : U → T ∗ M : p 7→ Lix (p).

We shall see that the above example of covector eld belong to the important class of exact dierential forms, for this scope we need the following basic denition.

Denition (dierential).

V → R,

locally dened on

M

at

p,

We dene the dierential at

p

of a smooth function

f :

as the following linear functional (covector)

dp f : Tp M → R : v 7→ v(f ). And we dene the dierential of

f

as the covector eld

df : V → T ∗ M sending each point

q

of the domain

V

to the form

dq f .

A covector eld

ω : V → T ∗M is named exact if

ω = df for some smooth function

f : V → R.

For example, the above covector eld Lix reveals none other than the exact dierential 1-form dxi , dierential of the i-th component of the chart x, selecting the i-th component of a vector eld v : U → T M, relative to the frame eld

∂/∂x : U → (T M )m : p 7→ ∂/∂x|p , induced by x on its domain U . 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.

119


5.4 Derivative versus dierentials Instead of dp f or (df )p , we also write df |p , for the value of the dierential 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 , 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, dened by [ f0 : M → Hom(Mp , Nf (p) ) : p 7→ f 0 (p), p∈M

for a map f between manifolds, where the linear operator f 0 (p) acts as it follows

fp0 (v)(g) = v(g ◦ f ), for every tangent vector v of M at p and for every real smooth function g locally dened at the point f (p) of N . Let us compare the two notions of derivative and dierential.

Proposition. domain manifold

M

If

f : M → R

is a smooth function, then, for every point

and every tangent vector

v

of the tangent space

Mp ,

p

of the

we obtain

fp0 (v) = (df )p (v)(d/dr)f (p) , where

r

represents the identity function of the real line.

Since the derivative fp0 (v) belongs to the tangent space Tf (p) R, then there exists a real number a such that fp0 (v) = a(d/dr)f (p) . Proof.

To evaluate the coecient a, we apply the above derivative to the identity function r. We obtain:

a =

a(d/dr)f (p) (r) =

=

fp0 (v)(r) =

=

v(r ◦ f ) =

=

v(f ) =

=

(df )p (v),

as we claimed. This proposition shows that - under the canonical identication of the tangent space Tf (p) R with the real line R, via the isomorphism (covector)

dp r : a(d/dr)f (p) 7→ a, induced by the identity function r of the real line R, the derivative fp0 corresponds to the dierential dp f .

120


5.5 Critical points of real functions In terms of the dierential df , we can dene the critical points of a real function dened on a manifold. We say that a point p of a manifold M is a critical point of a smooth function f : M → R if and only if (df )p = 0, that is, if and only if the dierential of f at p is the zero linear functional on Mp . We could show that local minimum points and local maximum points of a smooth function f :M →R are critical points.

5.6 Dierential of a vector function In this section, we generalize the concept of dierential for vector valued functions dened on dierentiable manifolds.

Denition (dierential of a vector function). smooth vector function

f : V → Rn ,

locally dened on

M

We dene the dierential at

at

p,

(n-covector)

dp f : Tp M → Rn : v 7→ v(f ) := (v(f j ))nj=1 = (dp f j (v))nj=1 . n-covector eld [ df : V → T ∗n M := Hom(Mp , Rn ) : q 7→ dq f,

And we dene the dierential of

f

as the

p∈M

sending each point

q

of the domain

V

to the operator

dq f .

We can associate, with the dierential 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 .

Denition (exact vector dierential form). ω : V → T ∗n M is named exact if

ω = df for some smooth function

f : V → Rn .

121

A

p

of a

as the following linear operator

n-covector

eld


Let us compare the two notions of derivative and dierential.

Proposition. manifold

M

f : M → Rn tangent vector v If

and every

fp0 (v) =

is a smooth function, then, for every point of the tangent space

n X

Mp ,

p

of the

we obtain

(df j )p (v)(∂/∂r)j |f (p) ,

j=1

where

r

represents the identity chart of the real Euclidean

n-space.

Since the derivative fp0 (v) belongs to the tangent space Tf (p) Rn , then there exists a real n-vector a such that Proof.

fp0 (v) =

n X

aj (∂/∂r)j |f (p) .

j=1

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

ak

=

n X

aj (∂/∂r)j |f (p) (rk ) =

j=1

= fp0 (v)(rk ) = = v(rk ◦ f ) = = v(f k ) = =

(df k )p (v),

for every k , as we claimed.

5.7 Change of covector frames Proposition (Transition matrix for covector frames). represent two coordinate charts on a manifold

dxi |p =

m X

M.

Suppose

(U, x) and (V, y)

Then, we see

(∂xi /∂y)j |p dy j |p ,

j=1

for every

p

in

U ∩V.

Consequently, we obtain

dxi =

m X

(∂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

i

of

x.

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

Xk := (∂/∂x)k |p ,

122


for every index k ; we see

dxi |p (Xk ) = Xk (xi ) = (∂/∂x)k |p (xi ) = δ ik and m X (∂xi /∂y)j |p dy j |p (Xk )

=

j=1

m X (∂xi /∂y)j |p Xk (y j ) = j=1

=

=

m X j=1 m X

(∂xi /∂y)j |p (∂/∂x)k |p (y j ) = (∂xi /∂y)j |p (∂y j /∂x)k |p =

j=1

=

[(∂x/∂y)|p (∂y/∂x)|p ]i k =

= δ ik , 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 , 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 .

5.8 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.

Denition (generalized contravariant vectors). M

and the

m-Cartesian

power

Vm

of a vector space

V.

Consider a smooth

We shall call a mapping

w : Ap → V m : w(x) = (w(x)i )m i=1 ,

123

m-manifold


a contravariant vector on

M

p

at

Vm

with values in

w(x)i =

if we see

m X (∂xi /∂y)j |p w(y)j , j=1

for every couple of charts

x, y

at

p,

on the manifold

M

and every index

i

of the Cartesian

power.

The rst 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.

Example (Albert Einstein). 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 dierentials ∂p x is an element (vector) of the m-Cartesian power (Mp∗ )m of the cotangent space Mp∗ . Consider, now the mapping w : Ap → (Mp∗ )m : w(x) = ∂p x, it reveals a contravariant vector on M at p with values in (Mp∗ )m . Indeed, we have

w(x)i

=

(∂p x)i =

= dp xi = m X = (∂xi /∂y)j |p dp y j = j=1

=

=

m X j=1 m X

(∂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 .

5.9 Generalized covariant vectors In a perfectly analogous way, we can dene the generalized covariant vectors.

Denition (generalized covariant vectors). and the

m-Cartesian

power

Vm

of a vector space

V.

Consider a smooth

m-manifold M

We shall call a mapping

w : Ap → V m : w(x) = (w(x)i )m i=1 , a covariant vector on

M

at

p

with values in

w(x)i =

m X

Vm

if we see

(∂y j /∂x)i |p w(y)j ,

j=1

for every couple of charts

x, y

at

p,

on the manifold

power.

124

M

and every index

i

of the Cartesian


Also in the present case we can propose a noble example. that

If (U, x) and (V, y) represent two coordinate charts on a manifold M at p, then, we saw

(∂/∂x)i |p =

m X (∂y j /∂x)i |p (∂/∂y)j |p , j=1

for every index i of x.

Theorem.

Consider the mapping

w : Ap → (Mp )m : w(x) = ∂/∂x|p , sending each local coordinate chart about a point

p.

p

of a manifold

M

into its induced tangent

reveals a generalized covariant vector on M at the point m in the Cartesian power (Mp ) (i.e, with component in the tangent space Mp .

frame at

Then,

Proof.

w

p

with values

Indeed, we have

w(x)i

= =

=

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

(∂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 x.

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