Marathe K. Topics in Physical Mathematics

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88 3 Manifolds

unity F = {f

| i ∈ I} subordinate to {U

| i ∈ I}. We observe that the

above deﬁnition of integral applies to



ω, ∀i ∈ I. Furthermore, one can

show that the number



i∈I



ω does not depend on the choices made for

A and F. Thus, we deﬁne the integral of ω on M by



ω =



i∈I



ω.

Using this deﬁnition of integration we now state the modern version of the

classical theorem of Stokes.

Theorem 3.4 (Stokes’ theorem) If (M, [ν]) is an oriented paracompact m-

dimensional manifold with boundary ∂M and ω is an (m − 1)-form on M

with compact support, then



dω =



∂M

∗

ω,

where i : ∂M → M is the canonical injection.

One often writes simply



∂M

ω instead of



∂M

∗

ω. The classical theorems

of Green (relating the line and surface integrals) and Gauss (relating the

surface and volume integrals) are special cases of the version of Stokes’ theo-

rem stated above. We observe that the deﬁnition of integral given above can

be extended to forms with compact support that are only continuous (not

necessarily smooth). Then, if f is a continuous function on M with compact

support and ω is a volume on M , we may deﬁne the integral of f with respect

to ω as



fω. It can be shown that in a suitable topology this deﬁnes a con-

tinuous functional on the space of continuous functions on M with compact

support. By the Riesz representation theorem, there exists a measure μ

the σ-algebra of Borel sets in M such that



fω =



fdμ

3.4 Pseudo-Riemannian Manifolds

In this section we take M to be a ﬁnite-dimensional manifold of dimension

m.Letg be a tensor ﬁeld of type (0, 2); we say that g is non-degenerate if,

for each p ∈ M , g(p) is non-degenerate, i.e.

g(p)(u, v)=0, ∀v ∈ T

M ⇒ u =0.

A tensor ﬁeld g ∈ T

M which is symmetric and non-degenerate is called

a pseudo-metric on M .Eachg(p) then deﬁnes an inner product on T

of signature (r, s). The numbers r, s are invariants of the pseudo-metric.

3.4 Pseudo-Riemannian Manifolds 89

The number r (resp., s) is the dimension of the positive eigenspace of g(p)

(resp., the negative eigenspace of g(p)) for each p ∈ M. The dimension of the

negative eigenspace of g is called the index of g. It is denoted by i

, i.e., the

index i

= s,wherer + s = m =dimM .Ifg is a pseudo-metric on M of

index s then we say that (M, g)isapseudo-Riemannian manifold of index

s. In local orthonormal coordinates the metric is usually expressed as



i=1

−



i=1

r+i

If s = 0, i.e., g(p) is positive deﬁnite ∀p ∈ M, then we say that (M,g)is

a Riemannian manifold. If s = 1, i.e., ∀p ∈ M the signature of g(p)is

(m − 1, 1), then we say that (M,g)isaLorentz manifold.Wenotethat

a pseudo-metric g induces an inner product on all tensor spaces, which

we also denote by g. In particular, if U is an open subset of M and

α =



<···<i

,...,i

∧···∧dx

,β=



<···<i

,...,i

∧···∧dx

are k-forms on U, the expression of g(α, β) is the following: Let g

−1

be the

inverse matrix of g = {g

} where g

= g(∂

,∂

)andletg

,...,i

,...,j

denote

the determinant of the k × k-matrix obtained taking the rows i

,...,i

and

the columns j

,...,j

of g

−1

.Then

g(α, β)=



<···<i

,...,i

where

,...,i



<···<j

,...,i

,...,j

Local expressions in pseudo-Riemmanian geometry simplify greatly when

they are referred to a coordinate chart with respect to which the matrix

of g is diagonal. Such coordinate charts always exist at each point of M and

the corresponding coordinates are called orthogonal coordinates.

There are several important diﬀerences in both the local and global prop-

erties of Riemannian and pseudo-Riemannian manifolds (see, for example,

Beem and Ehrlich [34] and O’Neill [308]). Until recently, most physical ap-

plications involved pseudo-Riemannian (in particular, Lorentz) manifolds.

However, the discovery of instantons and their possible role in quantum ﬁeld

theory and subsequent development of the so-called Euclidean gauge theo-

ries has led to extensive use of Riemannian geometry in physical applications.

It is these Euclidean gauge theories that have been most useful in applications

to geometric topology of low-dimensional manifolds.

If (M,g) is an oriented, pseudo-Riemannian manifold with orientation [ν],

then we deﬁne the metric volume form μ by μ = ν/|g(ν, ν)|

1/2

.Onan

oriented pseudo-Riemannian manifold (M,g)withmetricvolumeμ, we deﬁne

the Hodge star operator

90 3 Manifolds

∗ : Λ(M ) → Λ(M)

as follows. For β ∈ Λ

(M), 0 ≤ k ≤ m, ∗β ∈ Λ

m−k

(M) is the unique form

such that

α ∧∗β = g(α, β)μ, ∀α ∈ Λ

(M). (3.8)

One can show that

∗∗α =(−1)

+k(m−k)

α, ∀α ∈ Λ

(M), (3.9)

where i

is the index of the manifold M .Inparticular,wehave

∗μ =(−1)

, ∗1=μ.

The star operator is linear. Thus, for the local expression, it is enough to give

the expression for ∗(dx

∧···∧dx

). Taking into account that locally

μ = |det g|

1/2

∧···∧dx

we have

∗(dx

∧···∧dx

)=|det g|

1/2



<···<j

(−1)

+···+j

+k(k+1)/2

,...,j

,...,i

∧···∧



∧···∧



∧···∧dx

where



denotes suppression of dx

Deﬁnition 3.7 Let (M,g) be an m-dimensional oriented pseudo-

Riemannian manifold of index i

.Thecodiﬀerential δ is the linear map

δ : Λ(M) → Λ(M) of degree −1, which is deﬁned on Λ

(M) by

δ := (−1)

+mk+m+1

∗ d∗, (3.10)

where ∗ is the Hodge star operator.

We observe that if f ∈ Λ

(M)=F(M), then δf =0.Furthermore,asa

consequence of equations (3.7)and(3.9)wehave

:= δ ◦ δ =0. (3.11)

α =



<···<i

,...,i

∧···∧dx

is a k-form on a subset U of M, expressed in terms of orthogonal coordinates

), then the expression for δα is given by

δα = −|det g|

−1/2



<···<i

k−1

···g

k−1

3.4 Pseudo-Riemannian Manifolds 91

·∂

(α

j,i

,...,i

k−1

|det g|

1/2

)dx

∧···∧dx

k−1

If α ∈ Λ(M ), then α is said to be coclosed (resp., coexact)ifδα =0(resp.

α = δβ for some β ∈ Λ(M)). As a consequence of equation (3.11)every

coexact form is coclosed. The converse of this statement is, in general, true

only locally.

The exterior diﬀerential and codiﬀerential operators on a pseudo-

Riemannian manifold (M,g) are closely related to the classical diﬀerential

operators gradient, curl, divergence, and Laplacian, as we now explain. Let

V be a ﬁnite-dimensional, real vector space and f : V × V → R a bilinear

map. Then f induces a linear map f



: V → V

∗

deﬁned by



(u)(v)=f(u, v), ∀u, v ∈ V.

We note that f is non-degenerate if and only if f



is an isomorphism. In this

case, we deﬁne f



: V

∗

→ V by f



:= (f



)

−1

. Applying this pointwise to the

pseudo-Riemannian metric g we obtain the maps



: X(M) → Λ

(M),g



: Λ

(M) →X(M).

The map g





)issaidtolower (raise) indices. Locally, we have



∂

)=X

, where X

= X

and



(α

)=α

∂

, where α

= α

The operation of lowering or raising indices has an obvious extension to

tensors of arbitrary type. The gradient operator in a pseudo-Riemannian

manifold (M, g)isthemapgrad:F(M ) →X(M) deﬁned by

grad := g



◦ d.

If f is a function on an open subset U of M,wehave

grad f = g

∂f

∂x

∂

The divergence operator in an oriented pseudo-Riemannian manifold (M,g)

is the map div : X(M) →F(M) deﬁned by

div := −δ ◦ g



If X is a vector ﬁeld on an open subset U of M,wehave

div X = |det g|

−1/2

∂

(|det g|

1/2

92 3 Manifolds

The Hodge–de Rham operator Δ (also called the Laplacian)inanori-

ented pseudo-Riemannian manifold (M, g)isthemapΔ : Λ(M ) → Λ(M)

deﬁned by

Δ := dδ + δd. (3.12)

The operator Δ is at the basis of Hodge–de Rham theory of harmonic

forms. The classical Laplace–Beltrami operator or the Laplacian deﬁned as

div ◦grad equals −Δ on functions. The curl operator in an oriented pseudo-

Riemannian manifold (M,g) of dimension 3 is the map curl : X(M) →X(M)

curl := g



◦∗◦d ◦ g



If X is a vector ﬁeld on an open subset U of M,wehave

curl X =(−1)

|det g|

−1/2

{[δ

) −δ

)]δ

+[δ

) −δ

)]δ

+[δ

) −δ

)]δ

We observe that the well known classical results

A =gradφ ⇒ curl A =0 and B =curlA ⇒ div B =0,

where φ is a function and A, B are vector ﬁelds on R

are consequences of

the relation d

= 0. Note that, in view of the Poincar´e lemma, the converse

implications frequently used in physical theories to deﬁne the scalar and

vector potentials are valid only locally.

3.5 Symplectic Manifolds

Let M be an m-dimensional manifold and let ω ∈ Λ

(M). We say that ω is

non-degenerate if, ∀p ∈ M,

ω(p)(u, v)=0, ∀v ∈ T

M ⇒ u =0. (3.13)

If ω

(p)arethecomponentsofω(p) in a local coordinate system at p,then

condition (3.13)isequivalentto

det ω

(p) =0, ∀p ∈ M. (3.14)

Condition (3.14) together with the skew symmetry of ω implies that the di-

mension m must be even; i.e., m =2n. Then the condition (3.13)isequivalent

to the condition that ω

:= ω ∧ ω ∧···∧ω be a volume form on M, i.e.,

(p) =0, ∀p ∈ M.

3.5 Symplectic Manifolds 93

Recall that any 2-form α can be regarded as a bilinear map of T

M and hence

induces a linear map



(p):T

M → T

∗

as follows:



(p)(u)(v)=α(p)(u, v),

where u, v ∈ T

M. The non-degeneracy of ω ∈ Λ

(M) deﬁned above is then

equivalent to ω



being an isomorphism. Its inverse is then denoted by ω



Thus, a non-degenerate 2-form sets up an isomorphism between vector ﬁelds

and 1-forms. If X ∈X(M)thenwehaveω



(X)=i

ω.Letα, β be 1-forms;

the bracket of α and β is the 1-form

[α, β]=ω



([ω



(α),ω



(β)]).

We note that this form is well deﬁned for any non-degenerate 2-form ω.

Deﬁnition 3.8 A symplectic structure on a manifold M is a 2-form ω

that is non-degenerate and closed. A symplectic manifold is a pair (M,ω),

where ω is a symplectic structure on the manifold M.

Example 3.7 Let Q be an n-dimensional manifold. Let P = T

∗

Q be the

cotangent space of Q;thenP carries a natural symplectic structure ω deﬁned

as follows: Let θ be the 1-form on P deﬁned by

θ(α

)(X)=α

(ψ

∗

(X)), ∀α

∈ T

∗

Q, X ∈ T

where ψ is the canonical projection of P = T

∗

Q to Q. We deﬁne ω = −dθ.

The form θ is called the canonical 1-form and ω the canonical symplec-

tic structure on T

∗

Q. By deﬁnition, ω is exact and hence closed. Its non-

degeneracy follows from a local expression for ω in a special coordinate sys-

tem, called a canonical coordinate system, deﬁned as follows: Let {q

} be

local coordinates at p ∈ Q.Thenα

∈ P can be expressed as α

= p

.We

take Q

= q

◦ ψ, P

= p

◦ ψ as the canonical coordinates of α

∈ P .Using

these coordinates, we can express the canonical 1-form θ as

θ = P

It is customary to denote the canonical coordinates on P by the same let-

ters q

and from now on we follow this usage. The canonical symplectic

structure ω is given by

ω = −d(p

)=dq

∧ dp

From this expression it follows that

= dq

∧ dp

∧···∧dq

∧ dp

=0.

Further, the components of ω in this coordinate system are given by the matrix

94 3 Manifolds

(ω



0 I

−I 0



where I (resp., 0) denotes the n × n unit (resp., zero) matrix.

The above example is of fundamental importance in the theory of sym-

plectic manifolds in view of the following theorem, which asserts that, at least

locally, every symplectic manifold looks like T

∗

Theorem 3.5 (Darboux) Let ω be a non-degenerate 2-form on a 2n-

manifold M .Thenω is symplectic if and only if each p ∈ M has a local

coordinate neighborhood U with coordinates (q

,...,q

,...,p

) such that

= dq

∧ dp

Example 3.7 is also associated with the geometrical formulation of classical

Hamiltonian mechanics, where Q is the conﬁguration space of the mechan-

ical system and P is the corresponding phase space. We now explain this

formulation.

If (M, ω) is a symplectic manifold, then the charts guaranteed by Dar-

boux’s theorem are called symplectic charts and the corresponding co-

ordinates (q

) are called canonical coordinates.IfM = T

∗

Q,andω

is the canonical symplectic structure on it, then, in the physical literature,

the q

are called the canonical coordinates and the p

the corresponding

conjugate momenta. This terminology arises from the formulation of clas-

sical mechanics on T

∗

Q. We now indicate brieﬂy the relation of the classical

Hamilton’s equations with symplectic manifolds.

Let (M,ω) be a symplectic manifold. A vector ﬁeld X ∈X(M)is

called Hamiltonian (resp., locally Hamiltonian )ifω



(X) is exact (resp.,

closed). The set of all Hamiltonian (resp., locally Hamiltonian) vector ﬁelds

is denoted by HX(M )(resp.LHX(M)). If X ∈HX(M ), then there exists

an H ∈F(M) such that



(X)=dH. (3.15)

The function H is called a Hamiltonian corresponding to X.IfM is con-

nected, then any two Hamiltonians corresponding to X diﬀer by a constant.

Conversely, given any H ∈F(M )equation(3.15) deﬁnes the corresponding

Hamiltonian vector ﬁeld by ω



(dH), which is denoted by X

. The integral

curves of X

are said to represent the evolution of the classical mechani-

cal system speciﬁed by the Hamiltonian H. In a local canonical coordinate

system, these integral curves appear as solutions of the following system of

diﬀerential equations

∂H

∂p

, (3.16)

= −

∂H

∂q

. (3.17)

3.6 Lie Groups 95

This is the form of the classical Hamilton’s equations.Letf, g ∈F(M);

the Poisson bracket of f and g, denoted by {f, g}, is the function

{f,g} := ω(X

The Poisson bracket makes F(M ) into a Lie algebra. If X is a Hamiltonian

vector ﬁeld with ﬂow F

, then Hamilton’s equations can be expressed in the

form

(f ◦F

)={f ◦ F

,H}.

A Poisson structure on a manifold is a Lie algebra structure on F(M)that

is also a derivation in the ﬁrst argument of the Lie bracket. The Lie bracket

of f,g is also called the Poisson bracket and is denoted by {f,g}.Wecan

deﬁne a Hamiltonian vector ﬁeld on M corresponding to the Hamiltonian

function H by X

(g):={g,H}∀g ∈F(M). A manifold with a ﬁxed Poisson

structure is called a Poisson manifold. A symplectic manifold is a Poisson

manifold but the converse is not true. A Poisson structure induces a map of

∗

M to TM, but this map need not be invertible.

In view of Darboux’s theorem, a symplectic manifold is locally standard

(or rigid). Thus, topology of symplectic manifolds or symplectic topology is

essentially global. This has been an active area of research with close ties

to classical mechanics. Gromov’s deﬁnition of pseudo-holomorphic curves in

symplectic manifolds has provided a powerful tool for studying symplectic

topology. For symplectic 4-manifolds this led to the Gromov–Witten, or GW,

invariants. For a special class of 4-manifolds Taubes has shown that the GW

and SW (Seiberg–Witten) invariants contain equivalent information. There

is also a symplectic ﬁeld theory introduced by Eliashberg and Hofer with con-

nections to string theory and non-commutative geometry. We do not consider

these topics in this book. A general reference for symplectic topology is the

book by McDuﬀ and Salamon [283]. For quantum cohomology see the books

by Kock and Vainsencher [228], Manin [258], and McDuﬀ and Salamon [282].

3.6 Lie Groups

A manifold sometimes carries an additional mathematical structure that is

compatible with its diﬀerential structure. An important example of this is

furnished by a Lie group.

Deﬁnition 3.9 A Lie group G is a manifold that carries a compatible group

structure, i.e., the operations of multiplication and taking the inverse are

smooth. If G, H are Lie groups, a Lie group homomorphism f : G → H

is a smooth group homomorphism of G into H, i.e., f is smooth as a map of

manifolds and f (ab)=f(a)f(b), ∀a, b ∈ G. Isomorphism and automor-

phism of Lie groups are deﬁned similarly. Another useful concept is that

96 3 Manifolds

of an anti-homomorphism. A smooth map f : G → H is called an anti-

homomorphism of Lie groups if f(ab)=f(b)f(a), ∀a, b ∈ G.

We observe that it is enough to require that the multiplication be smooth

because this implies that the operation of taking the inverse is smooth. In-

version, i.e., the map ι : G → G deﬁned by a → a

−1

,a∈ G,isananti-

automorphism of the Lie group G. Furthermore, one can show that if G, H

are ﬁnite-dimensional Lie groups, then every continuous homomorphism of

G into H is smooth and hence is a Lie group homomorphism.

Example 3.8 Let F be a ﬁnite- (resp., inﬁnite-) dimensional Banach space.

The group of automorphisms (linear, continuous bijections) of F is a ﬁnite-

(resp., inﬁnite-) dimensional Lie group. Due to the fact that the locally com-

pact Hausdorﬀ manifolds are ﬁnite-dimensional, a locally compact Haudorﬀ

Lie group is ﬁnite-dimensional (see, for example Lang [245]).

If G is a locally compact group, there exists a unique (up to a multiplicative

constant factor) measure μ on the σ-algebra of Borel subsets of G that is left

invariant, i.e.,

μ(gU)=μ(U), ∀g ∈ G

and for all Borel subsets U of G. Such a measure is called a Haar measure

for G.IfG is compact, a Haar measure is also right invariant. Thus, ﬁnite-

dimensional Lie groups have a Haar measure and this measure is also right

invariant if the Lie group is compact.

A subgroup H of a Lie group is called a Lie subgroup if the natural

injection i : H → G is an immersion. If H is a closed subgroup of G,then

H is a submanifold of G and therefore is a Lie subgroup of G.Itiseasyto

check that the groups GL(n, R)andGL(n, C) deﬁned in Example 3.5 are

Lie groups. We now give an example of Lie subgroups.

Example 3.9 In Example 3.5, we introduced the smooth determinant map

det : M (n, R) → R.I

srestrictiontoGL(n, R) is also smooth. In fact,

det : GL(n, R) → R \ 0=GL(1, R)

is a Lie group homomorphism. It follows that the kernel of this homomor-

phism, det

−1

({1}) is a closed subgroup of GL(n, R). It is called the special

real linear group and is denoted by SL(n, R).Thus,

SL(n, R):={A ∈ GL(n, R) | det A =1}

is a Lie subgroup of GL(n, R). Analogously, one can show that the special

complex linear group

SL(n, C):={A ∈ GL(n, C) | det A =1}

is a closed Lie subgroup of GL(n, C).

The orthogonal group O(n) deﬁned by

3.6 Lie Groups 97

O(n):={A ∈ GL(n, R) | AA

=1},

where A

is the transpose of A, is the ﬁxed point set of the automorphism

of GL(n, R) deﬁned by A → (A

)

−1

. Hence, it is a closed Lie subgroup of

GL(n, R).Thespecial orthogonal group SO(n) is deﬁned by

SO(n):=O(n) ∩ SL(n, R).

The group SO(n) is usually referred to as the rotation group of the n-

dimensional Euclidean space R

. It is the connected component of the identity

in the orthogonal group. Similarly, the unitary group U(n) deﬁned by

U(n):={A ∈ GL(n, C) | AA

†

=1},

where A

†

denotes the conjugate transpose of A, is a closed Lie subgroup of

GL(n, C).Thespecial unitary group SU(n) is deﬁned by

SU(n):=U(n) ∩ SL(n, C).

Another deﬁnition of these groups is given in Example 3.12.

We give other important examples of Lie groups later.

A Lie group (left) action (or a G-action)ofaLiegroupG on a manifold

M is a smooth map

L : G × M → M, such that L

: M → M

deﬁned by L

(x)=L(g, x) (also denoted by gx) is a diﬀeomorphism of M

∀g ∈ G and

∀g

∈ G, L

= L

◦ L

and L

= id

, (3.18)

where e is the identity element of G.ThemapL induces the map

L : G →

Diﬀ(M ) deﬁned by g → L

. The conditions in equation (3.18)maybeex-

pressed by saying that the map

L : G → Diﬀ(M) is a group homomorphism.

In fact, this last statement is sometimes used as the deﬁnition of a Lie group

action. The orbit of x ∈ M under the G-action is the subset {gx | g ∈ G}

of M , also denoted by Gx. The set of the orbits of the G-action L on M is

denoted by M/L or by M/G when L is understood. A G-action on M is said

to be transitive if there is just one orbit; in this case we also say that G acts

transitively on M.Ifx ∈ M , then the isotropy group H

of the G-action

is deﬁned by

= {g ∈ G | gx = x}.

If G acts transitively on M then H

∼

, ∀x, y ∈ M ,andifH denotes

the isotropy subgroup of some ﬁxed point in M,thenM is diﬀeomorphic to

G/H. Such a diﬀerential manifold M is called a homogeneous space of G

or a G-homogeneous space. In particular, if H is a closed subgroup of G,then