Gliklikh Y.E. Global and Stochastic Analysis with Applications to Mathematical Physics

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6 1 Manifolds and Related Objects

the basis

∂

∂ ˙q

,...,

∂

∂ ˙q

in T

as described above. Now p : T

→ R

well-deﬁned by formula (1.2).

Consider a chart U

on M and denote by T U

the restriction of the tangent

bundle to U

.NotethatT U

can be presented as the direct product T U

× R

since every point of T U

can be described by 2n coordinates: n

coordinates of the point in U

and n coordinates of the vector with respect to

the basis

∂

∂q

. Such a presentation as a direct product is called a trivialization,

in the case under consideration, with respect to coordinates (q

,...,q

). If we

choose another system of coordinates in U

, we obtain another trivialization.

There exist some other types of trivializations, for example, by a ﬁeld of

orthonormal frames in a chart of a Riemannian manifold (see below).

Consider TU

as a chart in TM. The change of coordinates between T U

and T U

is given as a pair:

(ϕ

βα

,ϕ



βα

):U

× R

→ U

× R

(1.3)

where ϕ

βα

sends the coordinates of points with respect to U

into those of

the same points with respect to U

and ϕ



βα

sends coordinate columns of a

vector with respect to the basis

∂

∂q

into those of the same vector with respect

to the basis

∂

∂q



according to formula (1.1).

Note that if ϕ

βα

is a C

-map, ϕ



βα

is only a C

k−1

-map, i.e. TM is a C

k−1

manifold if M is C

. However, if M is C

∞

, TM is also C

∞

(recall that, unless

the contrary is stated, we assume our manifolds to be C

∞

Convention 1.3 In what follows we denote the points of a tangent bundle in

two diﬀerent ways: (m, X) as a point in TM and X

as a tangent vector to M

at m. Both symbols have the same meaning and can be used interchangeably;

we will favor one when it is more suitable that the other. Strictly speaking, this

notation makes proper sense only in charts but for the sake of convenience

we also use it in invariant language.

Deﬁnition 1.4. A vector ﬁeld on M is a map X : M → TM such that

πX = id, i.e., πX(m)=m for each m ∈ M.

Note that for a general mapping X : M → TM the value X(m) could

belong to any ﬁber but from the relation πX(m)=m it follows that X(m) ∈

Deﬁnition 1.5. We say that a vector ﬁeld X is continuous (smooth)ifthe

map X : M → TM is continuous (smooth, respectively).

Clearly the derivative of a smooth curve is a tangent vector. Let a smooth

vector ﬁeld X be given on M. The curve m(t) described by equation

˙m(t)=X(m(t)). (1.4)

is called the integral curve of X.

1.1 Manifolds, Vectors and Covectors. A Glossary 7

In a given chart, (1.4) can be considered as a diﬀerential equation in vec-

tor space. Hence, we can apply the theory of ordinary diﬀerential equations

to (1.4). In particular, since the right-hand side of (1.4) is smooth, we ob-

tain from the classical existence theorem that for any point m

∈ M there

exist a real number ε

> 0 and a unique solution m(t)of(1.4) with the

initial condition m(0) = m

that is well-deﬁned for t ∈ [0,ε

). In order

to investigate all solutions of (1.4) we denote by g

) the solution such

that g

)=m

. From the theorem on smooth dependence of a solution

on initial values and parameters it follows that g

):U → M is jointly

smooth in t and m

∈ M,whereU is some neighborhood of 0×M in R×M.

From the uniqueness theorem for solutions of ordinary diﬀerential equations

it follows that g

(·) is a diﬀeomorphism for all t such that g

(m) exists for all

m ∈ M.

Deﬁnition 1.6. g

(·) is called the general solution of (1.4)ortheﬂow of

vecotr ﬁeld X .

Let f : M → R be a real-valued function on M . Specify a point m ∈ M

and denote by m(t) the integral curve of X such that m(0) = m. Consider

the restriction of f onto m(t), i.e. the function f(m(t)). Note that f (m(t)) is

a real-valued function of a real argument.

Deﬁnition 1.7.

f(m(t))

|t=0

is called the derivative of f in the direction

of X at m. Having found the derivative of f along X at all points of M,we

obtain a new function on M, denoted by Xf, that is called the derivative of

f in the direction of X.

Note that applying the above construction to the derivative of f in the

direction of a vector ﬁeld

∂

∂q

, we obtain the partial derivative

∂f

∂q

.Thisis

the reason for denoting this vector ﬁeld by the partial derivative symbol.

Let X be a vector ﬁeld on M andinachartU

let it be presented in

coordinate form as X = X

∂

∂q

. Then we easily obtain the following formula

for Xf in U

Xf = X

∂f

∂q

(1.5)

Let X and Y be smooth vector ﬁelds on M.

Deﬁnition 1.8. The Lie bracket [X, Y ]ofX and Y is the vector ﬁeld on M

such that for any smooth real-valued function f on M its derivative along

[X, Y ] is given by the formula: [X, Y ]f = X(Yf) − Y (Xf).

Usually the deﬁnition of [X, Y ] is given in operator form as follows:

[X, Y ]=X ◦Y − Y ◦ X. (1.6)

By direct calculation one can easily show that the vector ﬁeld [X, Y ] exists,

is unique and, in local coordinates, is described by the formula

8 1 Manifolds and Related Objects

[X, Y ]=



∂Y

∂q

− Y

∂X

∂q



∂

∂q

. (1.7)

The Lie bracket is evidently skew-symmetric: [X, Y ]=−[Y,X].

Proposition 1.9 The Lie bracket satisﬁes the so-called Jacobi identity:

[X, [Y,Z]] + [Y,[Z, X]] + [Z, [X, Y ]] = 0. (1.8)

Consider a smooth mapping of manifolds F : M → N. Specify a point

m ∈ M and its image F (m) ∈ N. The mapping F generates a mapping

F : T

M → T

F (m)

N that in a chart at m and in a chart at F (m)is

described by the Jacobi matrix of F at m. We call d

F the diﬀerential of F

at m ∈ M.

Deﬁnition 1.10. The tangent mapping TF : TM → TN is deﬁned by the

formula

TF(m, X)=(F (m), d

F (X)).

So, the tangent mapping TF is deﬁned globally as a mapping of tangent

bundles, and the diﬀerential d

F of F at m is a restriction of TF to T

Note that for a smooth curve m(t)inM we have the relation

F (m(t)) = TF



m(t)



, (1.9)

which can be considered as a coordinate-free deﬁnition of TF.

If F sends M into a linear space, say, F : M → R

, the construction of

the diﬀerential can be modiﬁed so that it becomes a map from TM to R

We introduce it as the composition

dF = p ◦d

F : T

M → R

(1.10)

where p is deﬁned in (1.2). Note that dF can be applied to vectors from any

M, m ∈ M, so it is well-deﬁned on the entire tangent bundle TM.

Deﬁnition 1.11. A cotangent vector (which we also call a covector or 1-

form) b at m ∈ M is a linear functional on the tangent space T

M, i.e. a

linear map b : T

M → R. The set of all covectors at m is called the cotangent

space at m and is denoted by T

∗

By deﬁnition, T

∗

M is a linear space, dual to T

M. The total cotangent

bundle is denoted by T

∗

In every chart U

the coordinates (q

,...,q

) generate the basis dq

,...,

in every T

∗

M, dual to the basis

∂

∂q

,...,

∂

∂q

(i.e., dq

(

∂

∂q

)=δ

where δ

equals 1 if i = j and 0 otherwise). Thus every covector a in U

has coordinate

representation a

= a

(note that the indices of covectors are superscripts

1.1 Manifolds, Vectors and Covectors. A Glossary 9

while the indices for their coordinates are subscripts). The relation between

such presentations a

and a

in the charts U

and U

is given by the formula

= a

◦ (ϕ



βα

)

−1

, i.e., in coordinates a



∂q



. (1.11)

Note that in (1.11) the Jacobi matrix ϕ



βα

is applied to the row a

from

the right while in (1.1) it is applied to the column X

from the left. Having

transposed both sides of (1.11) we obtain (1.11) in the form similar to (1.1):

=[(ϕ



βα

)

−1

]

◦ a

. (1.12)

Proposition 1.12 The transformation rules (1.1) and (1.11) coincide if and

only if ϕ



βα

is an orthogonal matrix.

Indeed, if and only if ϕ



βα

is orthogonal, we have [(ϕ



βα

)

−1

]

= ϕ



βα

by a

characteristic property of orthogonal operators.

Recall that the velocity of a curve, in particular of the trajectory of a

mechanical particle, is a tangent vector. The momentum p of the trajectory

is introduced in elementary text books by the property that the inner product

of p and the velocity v is equal to the kinetic energy multiplied by 2:

p ·v =2K. (1.13)

Note that in fact p is a covector. Indeed the kinetic energy K is a scalar, i.e.

it takes the same value in all charts and coordinate systems. So, since v is

a vector (transforming under changes of coordinates by (1.1)), p ·v in (1.13)

can take the same value in all coordinate systems if and only if p transforms

by (1.11), i.e., if p is a covector.

Another physical example of a covector is force. Indeed, the inner product

of a force f and velocity v is a scalar known as the power N:

f · v = N.

As in (1.13), since v transforms according to (1.1)andN has the same value

in all coordinate systems, f must transform according to (1.11), i.e., it is a

covector.

Note that in elementary text books only motion in R

with orthonormal

coordinate systems is considered. Thus, only orthogonal coordinate changes

are used and so the vectors v and covectors p and f have the same transfor-

mation rules (see above). This is not the case for systems given on manifolds

with arbitrary coordinate systems.

As for TM, T

∗

M inherits a manifold structure from M: the charts on T

∗

are of the form T

∗

, which can be represented as U

× R

, and changes

of coordinates are of the form (ϕ

βα

, ((ϕ



βα

)

−1

)

)(formula(1.12)isinuse

instead of (1.1)). The points of T

∗

M we shall denote either by (m, b)(asa

point of T

∗

M) or, equivalently, as b

(covector b at the point m).

10 1 Manifolds and Related Objects

As for TM we denote by the symbol π : T

∗

M → M the projection which

sends (m, b) ∈ T

∗

M into m ∈ M.

Deﬁnition 1.13. A covector ﬁeld on M is a map b : M → T

∗

M such that

π ◦b = id (cf. Deﬁnition 1.4).

A covector ﬁeld is called continuous (smooth)ifb as a map of manifolds

is continuous (smooth, respectively).

Since R is a linear space, the construction of the diﬀerential df according

to (1.10) is well-deﬁned for any given real-valued function f : M → R. Here

we are to emphasize the following:

Proposition 1.14 df is a covector ﬁeld on M.

Indeed, by the construction, df is a map from TM into R which is linear

(coinciding with d

f)onanytangentspaceT

M. This means that at any

m ∈ M the map df is a linear functional on T

M, i.e. a covector.

In a given chart U

,df can be expressed in terms of coordinates with

respect to a basis dq

,... dq

. To do this we should calculate the Jacobi

matrix for f in coordinates q

,... q

. We then ﬁnd that it takes the form

(

∂f

∂q

,...,

∂f

∂q

) and hence

df =

∂f

∂q

. (1.14)

Remark 1.15. Note that in classical vector analysis (

∂f

∂q

,...,

∂f

∂q

)isknown

as the gradient of the function f while (1.14) has the form of a total diﬀer-

ential. We should point out that both formulas describe (in diﬀerent forms)

the object we have called the diﬀerential of f. The geometrically well-deﬁned

deﬁnition of gradient is given in the next chapter in such a way that in Eu-

clidean n-space it turns out to be a vector (i.e., a coordinate column, not a

row) with coordinates

∂f

∂q

Taking into account (1.14), we obtain for X = X

∂

∂q

that

df(X)=X

∂f

∂q

. (1.15)

Comparing (1.5) with (1.15) we obtain that the equality

Xf =df(X) (1.16)

holds for any X and any f.

Note that (1.16) is sometimes used as the deﬁnition of df.

Above we have described the notion of a tangent mapping of tangent

bundles generated by a smooth mapping of manifolds. A natural mapping of

a cotangent bundle is also generated but, unlike the tangent mapping, it sends

∗

N to T

∗

M.LetM and N be smooth ﬁnite-dimensional manifolds and let

1.2 Lie Groups and Lie Algebras 11

F : M → N be a smooth map. Specify a point m ∈ M and consider its image

F (m) ∈ N.Leta ∈ T

∗

F (m)

N be a covector at F (m). It can be mapped into

∗

M by the following procedure. Denote by T

∗

F (a) the covector in T

∗

such that its value on any vector X ∈ T

M is deﬁned by the relation

∗

F (a)(X)=a(TF(X)) (1.17)

where TF is the tangent mapping. By the construction of TF (see Deﬁnition

1.10), we get that TF(X) ∈ T

F (m)

N and so the value a(TF(X)) is well-

deﬁned.

Deﬁnition 1.16. The map T

∗

F : T

∗

N → T

∗

M is called the cotangent map-

ping of F : M → N or the pull-back.

1.2 Lie Groups and Lie Algebras

Deﬁnition 1.17. A manifold G is called a Lie group if there exists an alge-

braic operation • on G such that G is a group with respect to • and g

• g

is jointly smooth in g

∈ G as a map from G × G to G.

The ﬁrst examples of Lie groups are the space R

(which is obviously a

commutative Lie group with respect to addition) and the circle S

, i.e. the

set of complex numbers with unit modulus, with respect to multiplication

(also commutative).

It is clear that the n-dimensional torus T

= S

×···×S

(the cartesian

product of n copies of S

) is a Lie group. The group operation on T

is given

by coordinate-wise multiplication.

Remark 1.18. T

may also be described as the quotient space R

of R

with respect to the integral lattice Z

. This means that in R

the points whose

corresponding coordinates diﬀer from one other by an integer are considered

as equivalent and are pasted onto each other. It is clear that S

is obtained

from the corresponding coordinate axis in R

by factorization with respect

to integer points.

Let us turn to non-commutative groups. First we mention the three-

dimensional sphere S

(the set of points with unit modulus in the space

of quaternions) and the sphere S

(the set of points with unit modulus in the

space of octaves).

Consider the group of real invertible n × n matrices with respect to ma-

trix multiplication. We denote this group, called the general linear group,by

GL(n, R).

Theorem 1.19 GL(n, R) is a Lie group.

12 1 Manifolds and Related Objects

The closed set det

−1

(0) ∈ L(n, R) subdivides GL(n, R) into two connected

components: the matrices with positive determinants and those with negative

determinants. Since the determinant of a product equals the product of the

determinants, the product of two matrices with positive determinants has

positive determinant, i.e., this connected component is a Lie sub-group of

GL(n, R).

Another Lie sub-group of GL(n, R) is the group O(n) of orthogonal n ×

n matrices. Recall that a matrix A is orthogonal if its action on vectors

preserves the inner product in R

or, equivalently, if A

= A

−1

where A

is the transposed matrix. O(n) is a regular surface (i.e., a submanifold) in

GL(n, R).

Like GL(n, R), O(n) has two connected components: the orthogonal ma-

trices with determinant +1 and those with determinant −1. The former is a

Lie sub-group of O(n) (the product of matrices with unit determinant has

unit determinant). This group is called the special orthogonal group and is

denoted by SO(n). The matrices in SO(n) preserve the space orientation

while the matrices with determinant −1 reverse it. The latter set of matrices

is not a subgroup of O(n).

Deﬁnition 1.20. A left action of a Lie group G on a manifold M is deﬁned

if a certain C

∞

-map G ×M → M, denoted for g ∈ G and m ∈ M by gm,is

given such that the following hypotheses hold:

(i) for any g ∈ G the map g : M → M that sends m to gm is a diﬀeomor-

phism;

(ii) (g • h)m = g(hm)forg, h ∈ G, m ∈ M.

A right action of a Lie group G on a manifold M is the speciﬁcation of a

certain C

∞

-map from M ×G to G,forg ∈ G and m ∈ M , which satisﬁes (i)

and the following replacement of (ii):

(iii) (g • h)m = h(gm)forg, h ∈ G, m ∈ M.

When a right action is given, the notation mg for g ∈ G, m ∈ M is used

so that m(g •h)=(mg)h.

In what follows we shall denote the unit of a Lie group G by e.

For g ∈ G two special maps, the left translation L

: G → G and the

right translation R

: G → G, are deﬁned by the formulae L

h = g • h and

h = h •g, respectively, for any h ∈ G. From Deﬁnition 1.17 it follows that

both L

and R

are smooth maps of G.

Note that the tangent bundle TG is trivial, i.e., it can be presented as

direct product. Indeed, having taken a certain basis in T

G we can translate

it to T

G at each point g ∈ G by TL

(or TR

) and so obtain the presentation

of TG as G ×R

,wheren =dimG.

Deﬁnition 1.21. The vector ﬁeld on G obtained by left (right) translations

of a vector X ∈ T

G at all points of G is called the left-invariant (right-

invariant, respectively) vector ﬁeld, generated by X.

1.2 Lie Groups and Lie Algebras 13

It should be noted that one can imitate the construction of left- and right-

invariant vector ﬁelds and deﬁne left- and right-invariant Riemannian metrics

and more general tensors (see the deﬁnitions of these object below).

Proposition 1.22 Let

X and

Y be left-invariant (right-invariant) vector

ﬁelds on G generated by X, Y ∈ T

G. Then the vector ﬁeld [

Y ] is left-

invariant (right-invariant, respectively).

See, e.g., [26] for the proof of Proposition 1.22.

Denote by [X, Y ] the vector in T

G which generates [

Y ].

Deﬁnition 1.23. [X, Y ] is called the bracket of X, Y .

Proposition 1.24 The bracket in T

G introduced in Deﬁnition 1.23 satisﬁes

the Jacobi identity (1.8).

The assertion of Proposition 1.24 follows from Propositions 1.9 and 1.22.

Deﬁnition 1.25. A linear space on which an additional operation [·, ·]sat-

isfying the Jacobi identity is given is called a Lie algebra.

Thus, by Proposition 1.24, T

G has the structure of a Lie algebra.

Deﬁnition 1.26. The vector space T

G, equipped with the bracket deﬁned

in Deﬁnition 1.23, is called the Lie algebra of the Lie Group G.

We generally denote Lie groups by Latin capitals (say, G)andtheirLie

algebras by the corresponding lower case Fraktur characters (say, g).

It is known that every ﬁnite dimensional Lie algebra is the Lie algebra

of a certain Lie group, however diﬀerent Lie groups may have the same Lie

algebra. For example, a group and a subgroup of the same dimension have

the same algebra.

Let us present some examples of Lie algebras.

On every linear space one can introduce the trivial bracket deﬁned by the

equality [X, Y ] = 0 for every X and Y . Thus every linear space is a (trivial)

Lie algebra. It is clear that the algebras of the groups R

, S

and T

are

trivial, as well as the algebras of all commutative groups.

Consider Euclidean space R

together with the skew-symmetric vector

product operation. This operation is usually denoted by the bracket symbol,

a notation which we retain, i.e., for two vectors X, Y ∈ R

the symbol [X, Y ]

denotes their vector product. By direct calculation one can easily prove the

following:

Proposition 1.27 The vector product operation satisﬁes the Jacobi identity.

Thus, R

with the vector product is a Lie algebra.

The Lie algebra gl(n, R) is the set of all n×n matrices with linear addition

in the underlying vector space and with the bracket [A, B]=AB − BA

(commutator of matrices).

14 1 Manifolds and Related Objects

The groups O(n)andSO(n) generate the same Lie algebra (indeed, the

unit of O(n) is contained in the subgroup SO(n)). This algebra is denoted

by so(n). It consists of skew-symmetric n ×n matrices with the same bracket

as in gl(n, R)(so(n) is a Lie subalgebra of gl(n, R)).

A particular case, the algebra so(3), is of special interest to us because of

its use, below, in some examples of mechanical systems. The matrices from

so(3) have the form

⎛

⎝

0 ab

−a 0 c

−b −c 0

⎞

⎠

. (1.18)

Denote by Ψ : so(3) → R

the mapping that sends the matrix (1.18)tothe

vector

⎛

⎝

−a

−b

−c

⎞

⎠

. The next statement is obvious.

Proposition 1.28 Ψ is a linear isomorphism of so(3) to R

Proposition 1.29 For every A, B ∈ so(3) the operator Ψ sends the commu-

tator AB −BA to the vector product of Ψ(A) and Ψ(B) in R

We introduce on so(3) a bilinear form by the formula (A, B)=−

tr(AB).

This bilinear form is called the Killing form.

Proposition 1.30 −

tr(AB) equals the inner product of the vectors Ψ(A)

and Ψ(B) in R

Propositions 1.29 and 1.30 are proved by direct calculation with matrices.

Thus −

tr(AB) is an inner product in so(3) and so we have introduced

the structure of Euclidean space in so(3). From Propositions 1.29 and 1.30

we obtain:

Proposition 1.31 The linear isomorphism Ψ : so(3) → R

is an isomor-

phism of Lie algebras and of Euclidean spaces.

1.3 Fiber Bundles

Deﬁnition 1.32. A ﬁber bundle comprises the following ﬁve objects:

(i) a manifold M, called the base of the bundle;

(ii) a manifold E, called the total space of the bundle;

(iii) a manifold F , called the standard ﬁber of the bundle;

(iv) a Lie group G, called the structure group of the bundle;

(v) a smooth projection π : E → M, called the projection of the bundle,

and the following interrelations between these objects hold:

1.3 Fiber Bundles 15

1) a left action of G on F is given (see Deﬁnition 1.20);

2) for any chart U

in M the set π

−1

is homeomorphic to U

× F ;

3) for U

∩ U

= ∅ the “change of coordinates” from U

× F to U

× F

is given as the pair (ϕ

βα

(m)) where ϕ

βα

: U

αβ

→ U

αβ

is deﬁned

above and g

βα

(m):F → F is an element of G,smoothinm ∈ U

αβ

which is a diﬀeomorphism of F according to the deﬁnition of the left

action of G.

Usually we denote a ﬁber bundle by the symbol of its total space, say

E, and in this case E

denotes the ﬁber at the base point m (note that

= π

−1

(m) is homeomorphic to F ). In order to indicate the details of

a certain ﬁber bundle we may say that there is a bundle E over M or that

π : E → M is a ﬁber bundle. To denote the bundle with ﬁber F ,aconvenient

notation is F

Strictly speaking U

× F and U

× F are not charts, but U

× F and

× F contain the charts in E of the form U

× V

and U

× V

(where

and V

are charts in F ). Here, the restrictions of (ϕ

βα

) become real

changes of coordinates.

The simplest example of a ﬁber bundle is a trivial (or product) bundle

E = M ×F . Here all g

βα

are equal to the unique element e = id (unit) in G

(and so the group G may be reduced to its subgroup consisting of the unique

element e). The elements of any product bundle are scalars (see above). Recall

that under changes of coordinates in M, scalars are not transformed at all.

Note that according to Deﬁnition 1.32 every bundle over each chart U

presented as a trivial one by means of a certain diﬀeomorphism F

that is

called a trivialization (it is often said that over each chart the bundles are

trivial or that all bundles by means of Deﬁnition 1.32 are locally trivial).

It is important to understand from the very beginning that many diﬀerent

trivializations of a bundle may exist even over a speciﬁed chart.

Deﬁnition 1.33. A vector bundle is a ﬁber bundle where F = R

for a

certain k, G is the group GL(k, R) of non-degenerate linear operators in R

,or

a subgroup thereof, and the elements of G act on R

as linear automorphisms.

The product bundles with F = R

are examples of vector bundles. Two

more examples are the tangent and the cotangent bundles of a manifold.

Indeed, for the tangent bundle, M is the base, TM is the total space, the

ﬁber is R

(assuming that dim M = n), G = GL(n, R) with the natural

action on R

and g

αβ

(m)=ϕ



αβ

(m)(see(1.3)). For the cotangent bundle,

M is also the base, T

∗

M is the total space, the ﬁber is R

and G is also

GL(n, R) with the same natural action on R

but g

αβ

now takes the form

[(ϕ



αβ

)

−1

]

(see (1.12)).

Of course, in the inﬁnite-dimensional case Deﬁnition 1.33 is modiﬁed by

replacing R

with some Hilbert or Banach space and G = GL(n, R)bythe

group of bounded invertible linear operators.

Deﬁnition 1.34. A principal bundle is a ﬁber bundle where F = G.