Marathe K. Topics in Physical Mathematics

Подождите немного. Документ загружается.

78 3 Manifolds

The map φ

: U

→ R

such that

([(x

,...,x

n+1

)]) = (x

,...,x

n+1

)

is a chart, and one can show that the set {φ

,...,φ

n+1

} is an atlas for RP

The set RP

with the diﬀerential structure determined by this atlas is a

diﬀerential manifold also denoted by RP

and called the n-dimensional real

projective space. We observe that RP

is the manifold of 1-dimensional

vector subspaces (i.e., lines through the origin) in R

n+1

. Analogously, one

deﬁnes the n-dimensional complex projective space CP

.Thus,CP

the set of complex lines in C

n+1

, i.e.,

:= {[x] | x ∈ C

n+1

\{0}},

where [x]={λx | λ ∈ C \{0}}. A similar construction extends to the quater-

nionic case to deﬁne the quaternionic projective space HP

. Gener-

alizing the above construction of RP

one can deﬁne the manifold of p-

dimensional vector subspaces of R

n+p

. It is called the Grassmann mani-

fold of p-planes in R

n+p

and is denoted by G

n+p

). The construction of

Grassmann manifolds can be extended to the complex and quaternionic cases.

The construction of Grassmann manifolds given in the above example

generalizes to the case of p-dimensional subspaces of an inﬁnite-dimensional

Banach space F and gives examples of inﬁnite-dimensional diﬀerential man-

ifolds.

Example 3.4 Let M,N be diﬀerential manifolds and let (U, φ), (V,ψ) be

any two charts of M and N with ranges in the Banach spaces F and G,

respectively. Let us denote by φ × ψ the map

φ ×ψ : U ×V → φ(U) × ψ(V ) ⊂ F ⊕ G

deﬁned by (x, y) → (φ(x),ψ(y)).Thepair(U ×V, φ×ψ) is a chart of M ×N

and the collection of all the charts of this type is an atlas for M × N,which

determines a diﬀerential structure on M ×N. With this diﬀerential structure

M × N is a diﬀerential manifold, which is called the product manifold of

M and N an

d is also denoted by M ×N.

Analogously, one deﬁnes the product

manifold M

×M

×···×M

of any ﬁnite number of diﬀerential manifolds.

An example of a product manifold is the n-dimensional torus T

deﬁned by

:= S

×···×S

  

n times

Example 3.5 Let M (m, n; R) denote the set of all real m ×n matrices. The

map

φ : M (m, n; R) → R

3.2 Diﬀerential Manifolds 79

deﬁned by φ(A)=(a

,...,a

),forA =(a

) ∈ M (m, n; R), is a chart

on M (m, n; R) that induces a diﬀerential structure on M (m, n; R).Withthis

diﬀerential structure M (m, n; R) is a diﬀerential manifold of dimension mn.

Let M (n, R):=M(n, n; R) and let us denote by S(n, R) (resp., A(n, R))the

subset of M(n, R) of the symmetric (resp., anti-symmetric) n ×n matri-

ces. It is easy to show that S(n, R) and A(n, R) are submanifolds of M (n, R)

of dimension n(n +1)/2 and n(n − 1)/2,

respectively. Analogously, the set

M(m,

n; C) of complex m × n matrices is a diﬀerential manifold of dimen-

sion 2mn.LetM(n, C):=M(n, n; C) and let us denote by S(n, C) (resp.,

A(n, C)) the subset of M(n, C) of the Hermitian (resp., anti-Hermitian)

n ×n matrices. It is easy to show that S(n, C) and A(n, C) are submanifolds

of M (n, C) of dimension n

.Themap

det : M(n, R) → R,

which maps A ∈ M(n, R) to the determinant det A of A, is smooth. Hence

det

−1

(R \{0} ) is an open submanifold of M(n, R) denoted by GL(n, R).

The set GL(n, R) with matrix multiplication is a group. It is called the real

general linear group. One can show that matrix multiplication induces

a smooth map of GL(n, R) × GL(n, R) into GL(n, R). Similarly, one can

deﬁne the complex general linear group GL(n, C). The real and complex

general linear groups are examples of an important class of groups called Lie

groups, which are discussed in Section 3.6.

Let M be a diﬀerential manifold and (U, φ), (V, ψ) be two charts of M at

p ∈ M. The triples (φ, p, u), (ψ, p,v), for u, v ∈ F , are said to be equivalent if

D(ψ ◦ φ

−1

)(φ(p)) · u = v

where D is the derivative operator in a Banach space. This is an equivalence

relation between such triples. A tangent vector to M at p may be deﬁned

as an equivalence class [φ, p, u] of such triples. Alternatively, one can deﬁne a

tangent vector to M at p to be an equivalence class of smooth curves on M

passing through p and touching one another at p (see, for example, Abraham

et al. [2]). The set of tangent vectors at p is denoted by T

M;thissetisa

vector space isomorphic to F , called the tangent space to M at p.Theset

TM =



p∈M

can be given the structure of a smooth manifold. This manifold TM is called

the tangent space to M. A tangent vector [φ, p, u]atp may be identiﬁed

with the directional derivative u

, also denoted by u

, deﬁned by

: F(U ) → R,

such that

80 3 Manifolds

(f)=D(f ◦ φ

−1

)(φ(p)) · u.

If dim M = m and φ : q → (x

,...,x

)isachartatp, then the tangent

vectors to the coordinate curves at p are denoted by ∂/∂x

or simply by

∂/∂x

or ∂

,i=1,...,m. The set of vectors ∂

, i =1,...,m,formabasisof

the tangent space at p ∈ U.Asmoothmap

X : M → TM

is called a vector ﬁeld on M if X(p) ∈ T

M,∀p ∈ M . The set of all vector

ﬁelds on M is denoted by X(M). A vector ﬁeld X ∈X(M)canbeidentiﬁed

with a derivation of the algebra F(M), i.e., with the linear map F(M) →

F(M) deﬁned by f → Xf where (Xf)(p)=X(p)f satisfying the Leibnitz

property X(fg)=fXg+ gXf. On a coordinate chart, the vector ﬁeld X has

the local expression X = X

∂

and hence Xf has the local expression X =

∂

f. In the above local expressions we have used Einstein’s summation

convention (summation over repeated indices) and in what follows we shall

continue to use this convention. If X, Y ∈X(M) then, regarding them as

derivations of the algebra F(M), we can deﬁne the commutator or bracket

of X, Y by

[X, Y ]:=X ◦ Y − Y ◦ X.

It is easy to check that [X, Y ]isinX(M ). The local expression for the

bracket is given by

[X, Y ]=



∂Y

∂x

−

∂X

∂x



∂

where X = X

∂

and Y = Y

∂

.WenotethatX(M) under pointwise vector

space operations and the bilinear operation deﬁned by the bracket has the

structure of a Lie algebra as deﬁned in Chapter 1; i.e., the Lie bracket,

(X, Y ) → [X, Y ] satisﬁes the following conditions:

1. [X, Y ]=−[Y,X] (anticommutativity),

2. [X, [Y,Z]] + [Y,[Z, X]] + [Z, [X, Y ]] = 0 (Jacobi identity).

If V and W are Lie algebras, a linear map f : V → W is said to be a Lie

algebra homomorphism if

f([X, Y ]) = [f (X),f(Y )], ∀X, Y ∈ V.

The Lie algebra X(M ) is, in general, inﬁnite-dimensional.

If f ∈F(M,N), the tangent of f at p, denoted by T

f or f

∗

(p), is the

map

f : T

M → T

f(p)

such that

f(u

) · g = u

(g ◦ f ).

3.2 Diﬀerential Manifolds 81

The tangent of f, denoted by Tf,isthemapofTM to TN whose restriction

to T

M is T

Let E be a Banach space and F a closed subspace of E.WesaythatF is

complemented in E if there exists a closed subspace G of E such that

1. E = F + G, i.e., ∀u ∈ E,thereexistv ∈ F, w ∈ G such that u = v + w;

2. F ∩ G = {0}.

The space G in the above deﬁnition is called a complement of F in E.

We say that F ⊂ E splits in E,whenF is a closed complemented subspace

of E.WeobservethatifE is a Hilbert space, then any closed subspace of

E splits in E and that if E is ﬁnite-dimensional, then every subspace of E

splits in E.LetM,N be diﬀerential manifolds and f ∈F(M, N). We say

that f is a submersion at p ∈ M if T

f is surjective and Ker T

f splits

in T

M.Equivalently,f is a submersion at p ∈ M , if there exist charts

φ : U → φ(U)=A

× A

⊂ F ⊕ G and ψ : V → ψ(V )=A

at p and

f(p), respectively, such that f (U) ⊂ V and the representative ψ ◦ f ◦ φ

−1

of f in these charts is the projection onto the ﬁrst factor. If M and N are

ﬁnite-dimensional, then f is a submersion at p ∈ M if and only if T

f is

surjective, i.e., the rank of the linear map T

f is equal to dim N.Wesaythat

f is a submersion if it is a submersion at p, ∀p ∈ M . One can show that a

submersion is an open map. A point q ∈ N is said to be a regular value of

f if, ∀p ∈ f

−1

({q}), f is a submersion at p. By deﬁnition, points of N \f(M )

are regular. An important result is given by the following theorem.

Theorem 3.1 Let M,N be diﬀerential manifolds and q ∈ N aregularvalue

of f ∈F(M,N).Thenf

−1

({q}) is an (m − n)-dimensional submanifold of

M and, ∀p ∈ f

−1

({q}),T

−1

({q})) = Ker T

An equivalence relation ρ on a diﬀerential manifold M is said to be regular

if, for some diﬀerential structure on the set of equivalence classes M/ρ,the

canonical projection π : M → M/ρ is a submersion. Such a diﬀerential

structure, if it exists, is unique and, in this case, M/ρ with this diﬀerential

structure is called the quotient manifold of M by ρ. A useful result is given

by the next theorem.

Theorem 3.2 Let M,N be diﬀerential manifolds and let f ∈F(M, N) be a

submersion. Let us denote by ρ the equivalence relation on M deﬁned by f,

i.e., xρy if f(x)=f(y).Thenρ is regular and f(M ) is an open submanifold

of N diﬀeomorphic to the quotient manifold M/ρ.

Example 3.6 Let ρ be the equivalence relation of Example 3.3. Then, as a

set, RP

=(R

n+1

\{0})/ρ. One can easily verify that, with respect to the dif-

ferential structure deﬁned on RP

in Example 3.3, the canonical projection

π : R

n+1

\{0}→RP

is a submersion, and RP

is the quotient manifold

n+1

\{0})/ρ.Letμ

be the restriction to S

of the map π given above

and let ρ

be the equivalence relation induced by μ

on S

. Observe that

identiﬁes antipodal points x and −x. By Theorem 3.2, ρ

is regular and

82 3 Manifolds

/ρ

is diﬀeomorphic to RP

. Analogously, if ρ



(resp., ρ



)istheequiv-

alence relation on S

2n+1

(resp., S

4n+3

) deﬁning CP

(resp., HP

), then



identiﬁes all points of the circle S

= {λx ||λ| =1,λ ∈ C} (resp., ρ



identiﬁes all points of the sphere S

= {λx ||λ| =1,λ ∈ H}). Thus, CP

is diﬀeomorphic to S

2n+1

/ρ



and HP

is diﬀeomorphic to S

4n+3

/ρ



.From

this it follows that these projective spaces are connected and compact.

Let M, N be diﬀerential manifolds and f ∈F(M,N). We say that f is

an immersion at p ∈ M if T

f is injective and Im T

f splits in T

f(p)

Equivalently, f is an immersion at p ∈ M, if there exist charts φ : U →

φ(U)=A

⊂ F and ψ : V → ψ(V )=A

× A

⊂ F ⊕ G at p and f(p)

respectively, such that f(U ) ⊂ V and the representative ψ ◦ f ◦ φ

−1

of f in

these charts is the natural injection A

→ A

× A

.IfM and N are ﬁnite

dimensional then f is an immersion at p ∈ M if and only if T

f is injective

and this is true if and only if the rank of the linear map T

f is equal to

dim M .Wesaythatf is an immersion if it is an immersion at p, ∀p ∈ M.

An immersion f is, locally, a diﬀeomorphism onto a submanifold of N, but

f need not be injective. Even if f is an injective immersion, f (M) need not

be a submanifold of N. An injective immersion f is called an embedding if

f(M ) is a submanifold of N.

3.3 Tensors and Diﬀerential Forms

In this section we deﬁne various tensor spaces and spaces of diﬀerential forms

on a manifold and discuss some important operators acting on these spaces.

Let V denote a vector space and let V

∗

be the dual of V .Wedenoteby

(V ) (see Appendix C) the tensor space of type (r, s)overV , i.e.,

(V )=V ⊗ ...⊗ V



 

r times

⊗V

∗

⊗ ...⊗ V

∗

  

s times

Replacing T

M with various tensor spaces over T

M, a construction similar

to that of the tangent space, deﬁnes the tensor spaces on M . We deﬁne T

M =



p∈M

M),

where

M)=T

M ⊗ ...⊗ T



 

r times

⊗(T

∗

⊗ ...⊗ (T

∗

  

s times

The set T

M can be given the structure of a manifold and is called the

tensor space of type (r, s) of contravariant degree r and covariant degree s.

A smooth map

3.3 Tensors and Diﬀerential Forms 83

t : M → T

is called a tensor ﬁeld of type (r, s)onM if t(p) ∈ T

M), ∀p ∈ M.We

note that, if M is ﬁnite-dimensional, then T

M) may be identiﬁed with

a space of multilinear maps as follows. The element

⊗ ...⊗ u

⊗ α

⊗ ...⊗ α

∈ T

is identiﬁed with the multilinear map

∗

× ...× (T

∗

  

r times

×(T

M) × ...× (T



 

s times

→ R

deﬁned by

(β

,...,β

,...,v

) → β

) ...β

)α

) ...α

This map is extended by linearity to all of T

M). We note that, if dim M =

m,thendimT

M = m

r+s

.WeobservethatT

M = TM.ThespaceT

is denoted by T

∗

M and is called the cotangent space of M.Thespace

∗

(also denoted by T

∗

M) is called the cotangent space at p ∈ M .We

deﬁne T

M):=R. Hence, T

M := M × R. Thus, a tensor ﬁeld of type

(0, 0) can be identiﬁed with a function on M, i.e., an element of F(M). A

tensor ﬁeld of type (r, 0) (resp., (0,r)) is also called a contravariant (resp.,

covariant) tensor ﬁeld of degree r. A contravariant tensor ﬁeld t of degree r

is said to be symmetric if, ∀p ∈ M,

t(p)(β

,...,β

)=t(p)(β

σ(1)

,...,β

σ(r)

), ∀β

∈ (T

∗

, ∀σ ∈ S

where S

is the symmetric group on r letters. A contravariant tensor ﬁeld t

of degree r is said to be skew-symmetric if

t(p)(β

,...,β

)=sign(σ)t(p)(β

σ(1)

,...,β

σ(r)

∀p ∈ M, ∀β

∈ (T

∗

, ∀σ ∈ S

. Similar deﬁnitions of symmetry and skew

symmetry apply to covariant tensor ﬁelds.

We deﬁne A

M := M × R and denote by A

M, k ≥ 1, the manifold

M =



p∈M

M) ,

where

M)=T

∗

M ∧ ...∧ T

∗



 

k times

is the vector space of exterior k-forms on T

∗

M. The manifold A

M is called

the manifold of exterior k-forms on M .WenotethatA

M = T

∗

M.A

smooth map

84 3 Manifolds

α : M → A

is called a k-form on M if α(p) ∈ A

M), ∀p ∈ M.Thespaceofk-forms

on M is denoted by Λ

(M). We note, in particular, that Λ

(M)canbe

identiﬁed with F(M). We deﬁne the (graded) exterior algebra on M to be

the (graded) vector space Λ(M), deﬁned by

Λ(M)=

+∞



k=0

(M),

with the exterior product ∧ as multiplication.

If dim M = m and {e

,...,e

} is a basis for T

∗

M,then

∧···∧e

}

1≤i

<···<i

≤m

is a basis for A

M). Thus

dim A

M =





We note that Λ

(M)={0} for k>m.Anm-form ν is called a volume

form or simply volume on M if, ∀p ∈ M, ν(p) = 0. The manifold M is said

to be orientable if it admits a volume. Two volumes ν, ω on M are said to

be equivalent if ω = fν for some f ∈F(M ) such that f (p) > 0, ∀p ∈ M.

An orientation of an orientable manifold M is an equivalence class [ν]of

volumes on M .Apair(M, [ν]), where [ν] is an orientation of the manifold

M, is called an oriented manifold. Given an oriented manifold (M, [ν]), a

chart φ : q → (x

(q),x

(q),...,x

(q)) on U ⊂ M is said to be positively

oriented if dx

∧···∧dx

∈ [ν

]. Let (M, [ν]) be an oriented manifold

with boundary and let A = {φ

: U

→ φ

) ⊂ R

}|i ∈ I} be an atlas on

M of positively oriented charts. The orientation on δM induced by the forms

(−1)

∧···∧dx

m−1

relative to the charts of A is called the induced

orientation on δM.

Let f ∈F(M,N); then f induces the map

∗

: Λ(N ) → Λ(M),

called the pull-back map, deﬁned as follows. If α ∈ Λ

(N)=F(N)then

we deﬁne f

∗

α := α ◦ f ∈ Λ

(M)=F(M). If α ∈ Λ

(N), k ≥ 1, then

∗

α ∈ Λ

(M) is deﬁned by

∗

α)(p)(u

,...,u

):=α(f(p))(T

f(u

),...,T

f(u

)),

∀u

,...,u

∈ T

M. If f : M → N is a diﬀeomorphism and X ∈X(N), the

element f

∗

X of X(M ) deﬁned by

3.3 Tensors and Diﬀerential Forms 85

∗

X = Tf

−1

◦ X ◦ f

is called the pull-back of X by f .Thus,whenf is a diﬀeomorphism one can

extend the deﬁnition of the pull-back map f

∗

to the tensor ﬁelds of type

(r, s). In particular if X

,...,X

∈X(N)andα

,...,α

∈ Λ

(N), we have

∗

⊗ ...X

⊗ α

⊗ ...α

)=f

∗

⊗ ...⊗ f

∗

⊗ f

∗

⊗ ...⊗ f

∗

Given X ∈X(M)andp ∈ M,anintegral curve of X through p is a

smooth curve c : I → M ,whereI is an open interval around 0 ∈ R , c(0) = p,

and

˙c(t):=Tc(t, 1) = X(c(t)), ∀t ∈ I.

A local ﬂow of X at p ∈ M is a map

F : I ×U → M,

where U is an open neighborhood of p such that, ∀q ∈ U,themapF

: I → M

deﬁned by

: t → F (t, q)

is an integral curve of X through q. One can show that, ∀X ∈X(M )and

∀p ∈ M,alocalﬂowF : I ×U → M of X at p exists and the map F

deﬁned

(q)=F (t, q), ∀q ∈ U

is a diﬀeomorphism of U onto some open subset U

of M.

Deﬁnition 3.4 Let X ∈X(M) and let η be a tensor ﬁeld of type (r, s) on

M.TheLie derivative L

η of η with respect to X is the tensor ﬁeld of type

(r, s) deﬁned by

η)(p)=

[(F

∗

η)(p)]|

t=0

∀p ∈ M,whereF : I ×U → M is a local ﬂow of X at p. The above deﬁnition

also applies to diﬀerential forms; then η ∈ Λ

(M) implies L

η ∈ Λ

(M).

It can be shown that the deﬁnition of Lie derivative given above is inde-

pendent of the choice of a local ﬂow. We now give local expressions for the

Lie derivative. For this it is enough to give such an expression on an open

subset U of the Banach space F . These expressions will give the representa-

tive of the Lie derivative in every chart on M. We shall follow this practice

whenever we give local expressions. We consider only the ﬁnite-dimensional

case, i.e., F = R

; the inﬁnite-dimensional expressions are analogous. Let

x =(x

,...,x

) ∈ U and X = X

∂

(summation over repeated indices

is understood). For η = f ∈F(U), from the deﬁnition of Lie derivative it

follows that

f =

∂f

∂x

. (3.1)

86 3 Manifolds

For η = Y ∈X(U) the expression of the Lie derivative L

Y is

Y =



∂Y

∂x

−

∂X

∂x



∂

, (3.2)

where B = {∂

:= ∂/∂x

| k =1, 2,...,n} is the natural basis of the tangent

space at each point of U. From the deﬁnition of exterior diﬀerential operator

given below, it follows that the dual base of B is B



= {dx

| k =1, 2,...,n}.

Then, for η = α = α

∈ Λ

(U) the expression of the Lie derivative L

becomes

α =



∂α

∂x

∂X

∂x



. (3.3)

The Lie derivative is a tensor derivation, i.e., it is linear and satisﬁes the

Leibnitz product rule

(η

⊗ η

)=(L

) ⊗η

+ η

⊗ L

, (3.4)

where η

,η

are tensor ﬁelds. This fact and the local expressions given above

allow us to write down the local expression for the Lie derivative of any tensor

ﬁeld of type (r, s).

Deﬁnition 3.5 The exterior diﬀerential operator d of degree 1 on Λ(M)

is the map d : Λ(M) → Λ(M) deﬁned as follows: If f ∈ Λ

(M),thendf ∈

(M) is deﬁned by

df · X = Xf.

If ω ∈ Λ

(M), k>0,thendω is the (k +1)-form deﬁned by

dω(X

,...,X



i=0

(−1)

(ω(X

,...,

,...,X

))



0≤i<j≤k

(−1)

i+j

ω(L

,...,

,...,X

), (3.5)

where

denotes suppression of X

This deﬁnition implies that

d(α ∧β)=(dα) ∧ β +(−1)

α ∧ dβ, (3.6)

where α ∈ Λ

(M),β∈ Λ(M), and

:= d ◦ d =0. (3.7)

From the deﬁnition, it also follows that, if

α =



<···<i

,...,i

∧···∧dx

3.3 Tensors and Diﬀerential Forms 87

is a local expression of the k-form α on a subset U of M, then the (local)

expression of dα is given by

dα =



j=1



<···<i

(∂

,...,i

)dx

∧dx

∧···∧dx

If α ∈ Λ(M), then α is said to be closed (resp., exact)ifdα =0(resp.,

α = dβ for some β ∈ Λ(M )). As a consequence of equation (3.7)everyexact

form is closed. The converse of this statement is, in general, valid only locally,

i.e., if α ∈ Λ(M) is closed then, ∀p ∈ M , there exists a neighborhood U of

p such that α

is exact; i.e., there exists β ∈ Λ(U) such that α = dβ.This

statement is called the Poincar´e lemma. In the physical literature β is

referred to as a potential for α and one says that α is derived from a

potential.

Deﬁnition 3.6 Let X ∈X(M ) and α ∈ Λ(M).The(left)inner multipli-

cation i

α of α by X is deﬁned as follows. If α ∈ Λ

(M)=F(M),thenwe

deﬁne i

α =0.Ifα ∈ Λ

(M), k ≥ 1,theni

α ∈ Λ

k−1

(M) is deﬁned by

α(X

,...,X

k−1

)=α(X, X

,...,X

k−1

In the following theorem we collect some important properties of the op-

erators L

, d,andi

Theorem 3.3 Let M, N be diﬀerential manifolds, then we have

1. d(f

∗

α)=f

∗

(dα),f∈F(M, N),α∈ Λ(N).

2. [X, Y ]=L

Y, X,Y ∈X(M).

3. L

= i

◦ d + d ◦ i

on Λ(M ).

4. d ◦L

= L

◦ d on Λ(M).

5. i

[X,Y ]

= L

◦ i

− i

◦ L

on Λ(M).

Let (M, [ν]) be an oriented paracompact m-manifold. Let ω be an m-form

on M with compact support supp ω. We now deﬁne the integral of ω with

respect to the volume ν, denoted by



ωdν,orsimplyby



ω.First,letus

suppose that there exists a positively oriented chart (U, φ) with supp ω ⊂ U .

By the change of variables rule for integrals, we can show that the integral



∗

ωdx

...dx

does not depend on the choice of a positively oriented chart (U, φ) satisfying

the condition that supp ω ⊂ U. Thus, in this case we can deﬁne



ω :=



∗

ωdx

...dx

For an arbitrary m-form ω with compact support, we choose an atlas A =

{(U

,φ

) | i ∈ I} of positively oriented charts and a smooth partition of