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

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16 1 Manifolds and Related Objects

It should be pointed out that on a principal bundle E both the left and

the right actions of its structure group G on ﬁbers are well-deﬁned, since any

ﬁber of E is isomorphic to G. In particular, having speciﬁed g ∈ G,wecan

consider the right translation η ◦ g of each point η ∈ E. Thus we obtain the

right action of G on E (see Deﬁnition 1.20).

Every principal bundle generates the so-called associated bundle as follows.

Let E be a principal bundle with structure group G,andletF be a manifold

on which a left action of G is given. Consider the quotient of the direct product

E ×F with respect to the right action of G deﬁned by the formula (η, f)g =

(η ◦g, g

−1

f) (note that this is indeed a right action since (gh)

−1

= h

−1

Consider a chart U

on M. The restriction of E ×F on U

has the form U

G×F . The above-mentioned quotient space consists of the orbits of elements

of this product: for (m, h, f) ∈ U

×G ×F the orbit is the set (m, hg, g

−1

for all g ∈ G. Such an orbit is associated with the point (m, hf) ∈ U

× F .

Indeed, for any point of the orbit we have (m, (hg)(g

−1

f)) = (m, hf )andso

this association is well-deﬁned. Thus, over U

, the above quotient space is

presented in the form U

× F and the total quotient space is a bundle over

M with ﬁber F and structure group G where the g

αβ

are the same as in E.

Deﬁnition 1.35. The above quotient is called the bundle associated to E

with ﬁber F .

Notation 1.36 Denote by λ the mapping of E × F onto the total space

of the associated bundle that is the factorization sending the orbits to the

corresponding points in the quotient.

It should be noted that every non-principal ﬁber bundle is associated to

some principal bundle.

Let Q be a vector bundle with ﬁber R

and let it be an associated bundle

to a principal bundle E with G = GL(k,R). Observe that, clearly, any b ∈ E

canbeconsideredasaframeb = e

,...,e

in the ﬁber Q

πb

of Q through πb

and, consequently, as a linear map b : R

→ Q

πb

sending x =(x

,...,x

)to

bx = x

+ ···+ x

Deﬁnition 1.37. The frame bundle BM of a manifold M is a principal bun-

dle over M with G = GL(n, R)(n =dimM)andg

βα

= ϕ



βα

Note that TM and T

∗

M are bundles associated to BM, and so a point b ∈

BM may be regarded as a frame b = e

,...,e

in the tangent space T

πb

Thus b can also be considered as a linear mapping b : R

→ T

πb

M which sends

a vector x =(x

,...,x

) ∈ R

to the vector bx = x

+ ···+ x

∈ T

πb

Deﬁnition 1.38. A cross-section X of a ﬁber bundle E is a map X : M → E

such that π ◦X =id:M → M , i.e. X(m) ∈ E

for any m ∈ M.

Examples of cross-sections are vector ﬁelds (cross-sections of a tangent

bundle) and covector ﬁelds (cross-sections of a cotangent bundle). This im-

mediately follows from Deﬁnitions 1.38, 1.4 and 1.13. Every vector bundle E

1.3 Fiber Bundles 17

has a smooth cross-section, called the zero-section, which sends m ∈ M into

the origin of E

. Unlike vector bundles, a principal bundle has a global (i.e.,

deﬁned on the entire manifold M) continuous cross-section if and only if the

bundle is trivial.

Deﬁnition 1.39. If the tangent bundle of a manifold M is trivial, M is called

parallelizable or trivializable.

As mentioned above, all Lie groups are parallelizable by left- or right-

invariant vector ﬁelds. In particular, the n-dimensional torus is parallelizable.

Remark 1.40. A natural trivialization Φ of the tangent bundle T T

can be

described as a trivialization by coordinate frames as follows. Introduce on

the angle coordinate q

2π

where ϕ is an angle. We then obtain the

system of angle coordinates (q

,...,q

)onT

. At each point m ∈T

the

vectors

∂

∂q

,...,

∂

∂q

form an orthonormal frame in T

.For(X

∂

∂q

)

set Φ(X

∂

∂q

)

=(m, (X

,...,X

)) ∈T

× R

Deﬁnition 1.41. If in each tangent space T

M to a manifold M a k-

dimensional linear subspace β

is chosen that smoothly depends on m ∈ M

(in the sense described below), we say that a k-dimensional distribution β is

given on M.

Smooth dependence of β

on m means that in some neighborhood of each

point m ∈ M there are k smooth linearly independent vector ﬁelds X

,...,X

such that at each m



in this neighborhood the space β



is the linear span of

the vectors X



),...,X



If on M there exists a smooth vector ﬁeld nowhere equal to zero, the

straight lines spanned on the vectors of this ﬁeld form a 1-dimensional dis-

tribution.

Deﬁnition 1.42. A k-dimensional distribution β is said to be integrable if

for each point m there exists a k-dimensional submanifold M



 m such that

at every point m



∈ M



the property T





= β



is fulﬁlled. In this case

the manifold M



is called an integral manifold of the distribution β.

Every 1-dimensional distribution is integrable. Its integral manifolds are

integral curves of the vector ﬁeld on which the distribution is spanned. Dis-

tributions of dimension greater than 1 may not be integrable. There is a

necessary and suﬃcient condition for integrability that involves the notion of

involutory distributions.

Deﬁnition 1.43. A distribution β is called involutory ifforeverytwovector

ﬁelds X and Y on M such that X

∈ β

at each m ∈ M,theirLie

bracket [X, Y ] also belongs to the space of distributions at each point of M.

Theorem 1.44 (Frobenius’ Theorem) A distribution is integrable if and

only if it is involutory.

A proof of Theorem 1.44 can be found, for example, in [26, 172, 212].

18 1 Manifolds and Related Objects

1.4 Riemannian and Semi-Riemannian Metrics

Deﬁnition 1.45. We say that a Riemannian metric is given on a manifold

M if in each tangent space T

M, m ∈ M, a symmetric positive-deﬁnite

bilinear form ·, ·

is speciﬁed which depends smoothly on m (in the sense

deﬁned below). A manifold with a Riemannian metric is called a Riemannian

manifold.

When we say that ·, ·

depends smoothly on m we mean that for any

two smooth vector ﬁelds X and Y on M the real-valued function X



on M is smooth in m.

For a Riemannian manifold every tangent space T

M becomes a Euclidean

space with inner product ·, ·

In what follows, if it does not lead to confusion, we shall omit the point m

in the inner product notation. Indeed, X, Y  obviously has only one interpre-

tation: X and Y lie in the same tangent space and they are to be substituted

into the inner product of that space.

The ﬁrst example of a Riemannian metric is the so-called ﬁrst fundamental

form of a surface in Euclidean space. Recall that for X,Y ∈ T

M,whereM

is a surface in a Euclidean space E,thenumberI(X, Y ) (the value of the ﬁrst

fundamental form I at X and Y ) is deﬁned by the equality: I(X, Y )=(X, Y )

where (X,Y ) is the inner product in E. Note that, in spite of the fact that the

unique inner product in E is used to determine the inner products in tangent

spaces, the latter are still speciﬁc to their own spaces: we still cannot identify

diﬀerent tangent spaces and so cannot say whether the inner products are

the same.

An embedding i : M → N of a Riemannian manifold M into another

Riemannian manifold N is isometrical if X, Y 

= iX, iY 

for every pair

X, Y of vectors belonging to the same tangent space of M.Notethatanem-

bedded manifold with the ﬁrst fundamental form is isometrically embedded

into Euclidean space.

Theorem 1.46 (Nash [186]) Every n-dimensional Riemannian manifold can

be isometrically embedded into a Euclidean space R

with N =

n(n+1)(3n+

11).

In a chart U

the form ·, ·

can be described in terms of its matrix.

As is typical in linear algebra, we consider the coeﬃcients of ·, ·

with

respect to the basis

∂

∂q

deﬁned by the formula g

= 

∂

∂q

∂

∂q

.Notethat

the coeﬃcients g

depend on m ∈ U

and so they are real-valued functions

on U

The linearity of ·, · allows us to derive a coordinate formula for the inner

product of vectors involving the coeﬃcients g

. For vectors X = X

∂

∂q

and

Y = Y

∂

∂q

we obviously have X, Y  = g

. Note that this notation can

be applied both for vectors X and Y at a given point and for vector ﬁelds on

1.4 Riemannian and Semi-Riemannian Metrics 19

. In the latter case both the coeﬃcients g

and the coordinates X

and

are functions on U

One can easily prove that ·, ·

is smooth in m ∈ U

if and only if all of

the functions g

(m)aresmooth.

Using a Riemannian metric one can create natural analogs of many notions

of ordinary geometry in Euclidean space.

For a vector X ∈ T

M we deﬁne its norm X by the formula X =



X, X. For two vectors X,Y ∈ T

M denote by θ the angle between them.

Then cos θ =

X,Y 

X·Y 

.Letm(t), t ∈ [a, b], be a curve in M . Its length s is

deﬁned by the formula



 ˙m(t)dt. (1.19)

Note that all the notions introduced above, after embedding M isometrically

into a Euclidean space, coincide with the usual deﬁnitions of norm, angle and

length, respectively, in that space.

Let m

∈ M. Denote by ℵ the total set of piecewise smooth curves in

M connecting m

and m

Deﬁnition 1.47. The inﬁmum of the lengths of the curves in ℵ is called the

Riemannian (or internal) distance in M between m

and m

and is denoted

by ρ(m

Proposition 1.48 The distance ρ(·, ·) satisﬁes the axioms of a metric:

(i) ρ(m, m)=0and from ρ(m

)=0it follows that m

= m

;

(ii) ρ(m

)=ρ(m

) for every pair m

∈ M;

(iii) ρ(m

)+ρ(m

) ≥ ρ(m

) for any m

∈ M.

The proof of Proposition 1.48 can be found, for example, in [26].

Hence, from Proposition 1.48 it follows that a Riemannian manifold is a

metric space with respect to its Riemannian distance ρ.

Deﬁnition 1.49. The Riemannian metric is called complete if the metric

space M with distance ρ generated by the Riemannian metric is complete.

In this case one says that M is a complete Riemannian manifold .

Recall that a bilinear form is called non-degenerate if its matrix (g

)is

not degenerate, i.e., it is invertible. Since the bilinear form of a Riemannian

metric is positive-deﬁnite it follows that the form is non-degenerate.

Deﬁnition 1.50. We say that a semi-Riemannian metric is given on a man-

ifold M if in each tangent space T

M, m ∈ M, a symmetric non-degenerate

bilinear form ·, ·

is speciﬁed which depends smoothly on m. A manifold

with a semi-Riemannian metric is called a semi-Riemannian manifold.

So, a Riemannian metric is a particular case of a semi-Riemannian metric.

For a semi-Riemannian metric a scalar square X, X may be negative or

20 1 Manifolds and Related Objects

even zero for non-zero X, hence neither norm nor distance are well-deﬁned

on semi-Riemannian manifolds. Nevertheless, semi-Riemannian metrics ap-

pear naturally in many physical theories. The main example for us is the

appearance of semi-Riemannian metrics in relativity theory.

Notation 1.51 For a Riemannian or semi-Riemannian metric with matrix

) in a certain chart, we denote the inverse matrix (g

)

−1

by (g

), i.e.,

its coeﬃcients are denoted by g

Remark 1.52. One can easily show that (g

) is the matrix of an inner prod-

uct in the cotangent space at the corresponding point. Both the latter ﬁeld

of inner products in cotangent spaces and the Riemannian metric (i.e., the

ﬁeld of inner products in tangent spaces described by matrices (g

)) are

traditionally called “metric tensors”.

Real mechanical and physical objects can be described both as vectors

and as covectors which are said to be physically equivalent. The set up of a

physical problem usually involves a certain Riemannian or semi-Riemannian

metric on a manifold. These metrics determine the physical equivalence as

follows.

Deﬁnition 1.53. For X ∈ T

M the physically equivalent covector b

∈

∗

M is deﬁned by the relation b

(Y )=X, Y  for any Y ∈ T

For b ∈ T

∗

M the physically equivalent vector X

∈ T

M is deﬁned by the

relation b(Y )=X

,Y for any Y ∈ T

From Deﬁnition 1.53 it follows that for any X ∈ T

M the physically equiv-

alent covector b

exists and is unique. Moreover, one can easily calculate its

coordinates. Let X = X

∂

∂d

and Y = Y

∂

∂d

. Denote by X

the coordinates

of b

, i.e., b

= X

. Then, since b(Y )=X

and X,Y  = g

for

any coordinate column Y

, we obtain from the deﬁnition

= g

. (1.20)

Applying (1.20) to the matrix (g

), the inverse of the matrix of the Rie-

mannian metric, one can easily see that

= g

. (1.21)

Remark 1.54. Note that here we use the existence of the matrix (g

)

−1

, and do not use positive-deﬁniteness. Thus physical equivalence is

well-deﬁned for the general case of a semi-Riemannian metric, not only for a

Riemannian one.

Deﬁnition 1.55. The vector physically equivalent to the diﬀerential df of a

function f (see (1.14)) is called the gradient of f and is denoted by gradf.

1.5 Tensors 21

So, the gradient is determined by the relation df(X)=gradf,X for

any X. Comparing this formula with (1.16) we obtain the following equality

which holds for all X and f:

Xf =df(X)=gradf,X. (1.22)

Convention 1.56 In what follows we shall denote the coordinates of phys-

ically equivalent vectors and covectors by the same character, using upper

indices for vector coordinates and lower indices for covector coordinates as

in formulae (1.20) and (1.21).

In mechanics and physics (1.20)and(1.21) are usually called ‘formulae of

lifting and lowering indices’. The term “physical equivalence” is suggested in

[200].

Let M be a Riemannian manifold. If in some T

M, m ∈ U

,wehave

= δ



1ifi = j

0ifi = j

i.e., the basis

∂

∂q

consists of orthonormal vectors, the coordinates of physi-

cally equivalent vectors and covectors coincide. In particular, and this is true

only in this case, dq

and

∂

∂q

are physically equivalent. However, a coordi-

nate system satisfying g

= δ

can be created only in a Euclidean space.

In a general Riemannian manifold one can easily create a coordinate system

satisfying the above equality at a single point, but the relation will generally

not be satisﬁed at other points in its neighborhood. For semi-Riemannian

manifolds the situation is quite analogous, the only modiﬁcation being: if

∂

∂q

is an orthonormal basis, g

= ±1.

1.5 Tensors

Main deﬁnitions

In order to describe many physical and mechanical notions one needs to

use mathematical objects more general than vectors and covectors. These

objects are called tensors and, in common with vectors and covectors, they

are characterized by the rules for transformation of their components under

changes of coordinates. Moreover, vectors and covectors turn out to be trivial

particular cases of tensors.

Let m ∈ M and consider the Cartesian product of r copies of T

∗

M and s

copies of T

Deﬁnition 1.57. A tensor of type (r, s)(or(r, s)-tensor)atthepointm is

a polylinear form on the above Cartesian product.

22 1 Manifolds and Related Objects

This means that a tensor T of type (r, s) is a real-valued function, linear

in all its arguments, such that for any collection of r covectors a,b,...,c and

s vectors X,Y,...,Z the real value T(a,b,...,c,X,Y,...,Z) is deﬁned.

A tensor ﬁeld on M is deﬁned if we associate to each point of M atensor.

Deﬁnition 1.58. The total number of arguments in a tensor T is called its

valency. The number of covector arguments is called the contravariant rank

of T and the number of vector arguments is called the covariant rank of T.

Scalars are tensors of valency 0. Tensors of valency 1 are vectors and

covectors. By deﬁnition, a vector is a tensor of contravariant rank 1 and a

covector is a tensor of covariant rank 1.

There exists a construction allowing one to create tensors of higher va-

lency from vectors and covectors. Consider vectors E

,...,E

and covectors

,...,p

. Deﬁne the tensor T = E

⊗···⊗E

⊗ p

⊗···⊗p

of type (r, s)

by the equality

⊗···⊗E

⊗ p

⊗···⊗p

,...,a

,...,X

)

= E

) ...E

) ...p

) (1.23)

for every multiplicity of covectors a

,...,a

and vectors X

,...,X

Deﬁnition 1.59. A tensor of the form (1.23) is called the tensor product of

vectors E

,..., E

and covectors p

,...,p

. Tensors of this type are called

elementary.

The space of tensors of type (r, s) at a point m is obviously a linear space.

It is clear that the elementary tensors

∂

∂q

⊗···⊗

∂

∂q

⊗ dq

⊗···⊗dq

, (1.24)

where the indices i

,...,i

,...,j

take all values from 1 to n (we assume

an n-dimensional manifold M), form a basis for this space.

For any (r, s)-tensor T, its coordinates with respect to the basis (1.24)are

denoted by T

...i

...j

so that

T = T

...i

...j

∂

∂q

⊗···⊗

∂

∂q

⊗ dq

⊗···⊗dq

. (1.25)

In order to avoid any confusion (we usually assume that coordinates appear

with respect to a basis in T

∗

M or in T

M), we call T

...i

...j

the components

of the tensor T.

Note that, for the sake of simplicity, the summands of (1.25) are ordered so

that all

∂

∂q

precede all dq

in the tensor products (and consequently, in the

components all terms with upper indices precede all terms with lower indices).

In arbitrary tensors the factors of types

∂

∂q

and dq

(and component terms

with upper and lower indices) may appear in any order.

1.5 Tensors 23

Taking into account formulae (1.1)and(1.11), one can easily derive the

following formula for the transformation of components under a change of

coordinates:



...i



...j



...



...



...i

...j

(1.26)

Note that all upper indices are transformed as in (1.1) and all lower indices

as in (1.11).

Formula (1.26) is the main characteristic of tensors. The above deﬁnition

of a tensor as a polylinear form is convenient only from the general point of

view. For concrete tensors, the fact that they can be presented as polylinear

forms may not be important for a given physical problem. On the other hand,

transformation rule (1.26) distinguishes tensors from all other objects.

In analogy with the constructions of tangent and cotangent bundles we

deﬁne the (r, s)-tensor bundle over M as the set of all (r, s)-tensors at all

points of M and deﬁne a smooth manifold structure on it as follows. The

sets of tensors over the charts U

and U

in M play the roles of charts in

our bundle (note that using (1.25) one can easily see that these new charts

are presented as direct products) and the changes of coordinates are given in

the form (ϕ

βα

)whereϕ

βα

is the change of coordinate in M and g

βα

deﬁned by (1.26). Now one can easily give the deﬁnition of an (r, s)-tensor

ﬁeld as a cross-section of a tensor bundle according to Deﬁnition 1.38 (i.e.,

in analogy with Deﬁnitions 1.4 and 1.13).

It is clear that a Riemannian metric is an example of a (0, 2)-tensor, i.e.,

in a chart, ·, · = g

⊗ dq

. The matrix (g

), the inverse of the matrix

) of a Riemannian metric in a certain coordinate system, is an example of

a(2, 0)-tensor, i.e., in a chart this metric tensor takes the form g

∂

∂q

⊗

∂

∂q

Thus, in any T

∗

M this matrix describes the inner product of covectors, dual

to the inner product on vectors in T

M (the Riemannian inner product)

with respect to physical equivalence.

It appears that (1, 1)-tensors are linear operators. Indeed, consider an ele-

mentary tensor E ⊗p at some m ∈ M and substitute into p a certain vector

X ∈ T

M.ThenE ⊗ p(·,X)=p(X)E ∈ T

M is vector linearly depen-

dent on X, i.e. E ⊗ p : T

M → T

M is a linear operator. Thus the tensor

T = a

∂

∂q

⊗ dq

is the linear operator in T

M with matrix (a

Note that we can substitute a covector a ∈ T

∗

M into E in E ⊗p so that

E ⊗p(a, ·)=E(a)p ∈ T

∗

M, i.e. E ⊗p can be considered as a linear operator

acting on T

∗

M. Hence a

∂

∂q

⊗dq

can also be considered as a linear operator

on T

∗

M.Thisoperatorisknownasthedual (or conjugate)operatortothe

operator on T

M, mentioned above, with the same matrix (a

Operations with tensors

The set of all (r, s)-tensors at a point m ∈ M form a linear space, i.e., addition

and multiplication (by a real number) are well-deﬁned. From this it follows

24 1 Manifolds and Related Objects

that for tensor ﬁelds the operations of addition and of multiplication by a

real-valued function are also well-deﬁned.

Further operations will be ﬁrst determined for elementary tensors. Since

every tensor is a sum of elementary tensors (see expansion (1.25)), we can

easily extend the operations from elementary to all tensors.

The tensor product of an (r, s)-tensor A and a (k, l)-tensor B creates a new

(r + k, s+ l)-tensor A⊗B as follows. For A = E

⊗···⊗E

⊗p

⊗···⊗p

and

B =

⊗···⊗

⊗ ¯p

⊗···⊗¯p

we have A ⊗B = E

⊗···⊗E

⊗p

⊗···⊗

⊗

⊗···⊗

⊗ ¯p

⊗···⊗¯p

. For arbitrary tensors T and

T, expanded

according to (1.25), we multiply by this rule each summand of T with each

summand of

T and sum up all the products to get the expansion of T ⊗

This means that the set of components for T ⊗

T is obtained by taking the

product of all components of T with all components of

A contraction (or trace) transforms an (r, s)-tensor into an (r − 1,s− 1)-

tensor as follows. For A = E

⊗···⊗E

⊗ p

⊗···⊗p

its contraction by

l − th lower and v − th upper factors is the tensor of the form

⊗···⊗E

l−1

⊗ E

l+1

...E

⊗ p

⊗···⊗p

v−1

⊗ p

v+1

⊗···⊗p

Applying this rule to a general tensor T with expansion (1.25) and taking

into account that dq

(

∂

∂q

)=δ

, we see that the contraction causes a lot of

summands in (1.25) to vanish. The remaining non-zero summands have the

form T

...i

l−1

l+1

...i

...j

v−1

v+1

...j

(the sum with respect to k =1,...,n).

For example, consider the contraction of a (1, 1)-tensor (linear operator)

T = a

∂

∂q

⊗ dq

(see above). Evidently we get trT = a

, the ordinary trace

of the matrix of the linear operator T.

Physically equivalent tensors

Let M be a Riemannian manifold. For an elementary tensor we deﬁne a

physically equivalent tensor by replacing a certain vector (or covector) in the

corresponding tensor product by the physically equivalent covector (vector,

respectively).

When transforming an arbitrary tensor with expansion (1.25) into a phys-

ically equivalent one, we should remember that generally speaking

∂

∂q

is not

physically equivalent to dq

. This is why, on replacing a certain

∂

∂q

(of dq

)

by the physically equivalent object, we should create the expansion (1.25)for

the obtained tensor. Taking into account formulae (1.20)and(1.21), one can

easily derive the corresponding formulae for arbitrary tensors. For the sake of

simplicity we present them for tensors with three indices; the general formu-

lae are analogous. Usually the components of physically equivalent tensors

are denoted by the same character, changing only the location of indices. So,

1.6 Diﬀerential Forms and Polyvectors 25

= g

ljk

;

= g

ilk

;

= g

uvk

and so on.

Remark 1.60. Note that the two versions of a metric tensor, i.e., matrices

)and(g

) (see Remark 1.52), are physically equivalent to each other.

Symmetric tensors

We say that an (r, 0)- or (0,s)-tensor is symmetric or skew-symmetric if

the polylinear form describing the tensor is respectively symmetric or skew-

symmetric. A symmetric form is a form whose value does not depend on

the order of its arguments while the sign of a skew-symmetric form changes

whenever two of its arguments are interchanged.

We consider (0,s)-tensors since the case of (r, 0)-tensors is analogous. One

can easily see that for a symmetric tensor, if the basis tensors in expansion

(1.25) diﬀer only by the order of their factors, the corresponding components

are equal. In order to simplify the notation we introduce the notion of a

symmetric tensor product:

···dq



⊗···⊗dq

where the summands dq

⊗···⊗dq

diﬀer only by the order of factors (it

is obvious that the number of such summands is s!). Thus, any symmetric

tensor has the expansion:

T = T

...i

···dq

where the indices i

, ..., i

are written in increasing order. One can also

easily show that the tensors dq

···dq

form a basis in the space of all

symmetric (0,s)-tensors.

1.6 Diﬀerential Forms and Polyvectors

A skew-symmetric (0,s)-tensor given at a point on a manifold is called an

exterior s-form and the corresponding tensor ﬁeld is called a diﬀerential s-

form; the valency of these tensors is called the degree of the form.

Skew-symmetric (r, 0)-tensors are called r-vectors. A vector ﬁeld in which

all the tensors are r-vectors is sometimes called an r-vector ﬁeld.Ifone

needn’t indicate the valency, we simply refer to an exterior (diﬀerential) form