Eschrig H. Topology and Geometry for Physics

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Chapter 4

Tensor Fields

4.1 Tensor Algebras

Let V and V

be two arbitrary real vector spaces of any ﬁnite dimension each.

Consider the product vector space V  V

consisting of all ordered pairs ðv; v

Þ; v 2

V; v

2 V

: For instance, given v 2 V; v

2 V

and k; l 2 R; ðkv; lv

Þ and ðlv; kv

are two different vectors of V  V

: Let W be the free real vector space generated

by V  V

; that is, the vector space consisting of all real linear combinations of

vectors out of V  V

: For instance, 2kðv; v

Þ; ðkv; v

Þþðv; kv

Þ are two more of

different vectors of W: Let I be the subspace of W generated by all vectors of the

forms

ðv

þ v

; v

Þðv

; v

Þðv

; v

Þðv; v

þ v

Þðv; v

ðkv; v

Þkðv; v

Þðv; kv

Þkðv; v

Þ:

The tensor product V  V

of the two vector spaces V and V

is the quotient space

W=I: Its elements are linear combinations of the tensor products v v

of vectors

v 2 V; v

2 V

(image of the canonical mapping of V  V

to V  V

) with the

properties

ðv

þ v

Þv

¼ v

 v

þ v

 v

;

v ðv

þ v

Þ¼v  v

þ v v

;

kðv v

Þ¼ðkvÞv

¼ v ðkv

Þ:

ð4:1Þ

Besides the elements of the form v  v

; the space V  V

contains all their linear

combinations. However, the canonical mapping / : V  V

! V  V

ðv; v

Þ7!v  v

of ordered pairs ðv; v

Þ to their equivalence classes v v

of the

quotient space formation is a universal bilinear mapping in the following sense: If

W is any vector space and w : V  V

! W is any bilinear mapping, then there is a

unique linear mapping w

: V  V

! W so that w ¼ w

 /: Up to isomorphisms,

/ is uniquely determined by this universality property.

H. Eschrig, Topology and Geometry for Physics, Lecture Notes in Physics, 822,

DOI: 10.1007/978-3-642-14700-5_4, Ó Springer-Verlag Berlin Heidelberg 2011

There are unique (obvious, canonical) isomorphisms between V  V

and V



V and between V ðV

 V

Þ and ðV  V

ÞV

: Therefore it makes sense to

write V  V

 V

and accordingly for more factors, and the order of factors can

be ﬁxed by convention in these product constructions. (This does of course not

mean that v

 v

þ v

 v

and v

 v

þ v

 v

are equal; the second expression

as a linear combination of elements of two different spaces is even not deﬁned, if

V and V

are different, and is different from the ﬁrst expression, if V ¼ V

However, the ﬁrst expression and v

ðv

þ v

Þ are conjugate by the canonical

isomorphism.)

Let fe

; ...; e

g and fe

; ...; e

g be bases of the vector spaces V and V

: Then

 e

g; i ¼ 1; ...; n

; j ¼ 1; ...; n

is a base of V  V

All these are simple statements which can be proved as an exercise (cf. for

instance [1]).

Let now V be any ﬁnite-dimensional real vector space and let V



be its dual.

Then the tensor space V

r;s

of type ðr; sÞ is

r;s

¼ V V

|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

r copies

 V



V



|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

s copies

: ð4:2Þ

A base in V

r;s

e

 f

f

g; ð4:3Þ

where the e

form a base of V and the f

form a base in V



conveniently chosen

dual to that of V : hf

; e

i¼d

: Additionally one deﬁnes V

0;0

¼ R: If V is

n-dimensional, then the tensor space V

r; s

is an n

rþs

-dimensional vector space. A

general element of V

r;s

is the tensor

t ¼

...i

...j

e

 f

f

; ð4:4Þ

where the summation runs from 1 to n over all indices which appear twice on the

right hand expression, once as subscript and once as superscript. Further on, the

summation sign will be omitted in tensor calculus, but not in exterior calculus for

reasons becoming evident below, and Einstein’s summation convention will be

used which means the just described summation over pairs of indices always

understood.

Now, the direct sum

TðVÞ¼

r;s 0

r;s

; V

0;0

¼ R; ð4:5Þ

is called the tensor algebra of V: It is an associative but non-commutative (see

remark in parentheses above on this page) graded (by r; s) algebra with unit. If

v

 w



w



2 V

and v

v

 w

1



s

2 V

; then their tensor product is deﬁned as v

v

 v



 w

1

w

s

 w

1

w

s

2 V

þr

þs

where the order of factors

98 4 Tensor Fields

belongs to the deﬁnition of the product in the algebra (see above). The product of

the two considered tensors as factors in inverse order is v

v

 v



v

 w

1

w

s

 w

1

w

s

2 V

þr

þs

: Tensors in some

r; s

are called homogeneous of degree ðr; sÞ: If they are single tensor products of

vectors and covectors as in the examples just considered, then they are called

decomposable.

A change of the base and dual base,

¼ w

; f

¼ðw

1

; w

ðw

1

¼ d

; ð4:6Þ

with a regular n n transformation matrix w; which should leave the tensor (4.4)

unaffected, results in a transformation of the tensor components according to

...i

...j

¼ w

w

...k

...l

ðw

1

ðw

1

: ð4:7Þ

Hence, a tensor transforms like a contravariant vector with respect to its upper

indices and like a covariant vector with respect to its lower indices. The tensor

product of the tensors t 2 V

r;s

and t

2 V

has components

ðt  t

...i

rþr

...j

sþs

¼ t

...i

...j

rþ1

...i

rþr

sþ1

...j

sþs

; ð4:8Þ

while the reversed order of the two factors leads to the reversed arrangement of the

index groups in the tensor product and hence in general to a different result. Two

tensors are equal if they have the same components with the order of indices

observed.

Consider a decomposable tensor t of degree ðr; sÞ and two integers p; 1 p r;

and q; 1 q s: The tensor contraction C

p;q

: V

r; s

! V

r1;s1

is deﬁned as

p;q

ðtÞ¼C

p;q

ðv

v

 w

1

w

s

¼hv

; w

q

v

p1

 v

pþ1

v

 w

1

w

q1

 w

qþ1

w

s

; ð4:9Þ

and for an arbitrary homogeneous tensor of degree ðr; sÞ it is deﬁned by linear

continuation. For an arbitrary homogeneous tensor of degree ðr; sÞ one has in

components (summation over k)

p;q

ðtÞ

...i

r1

...j

s1

¼ t

...i

p1

...i

r1

...j

q1

...j

s1

: ð4:10Þ

There are various interrelations of tensors with mappings. First, every homo-

geneous tensor of degree ðr; sÞ may be considered as a multilinear mapping

t : V



V



|ﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄﬄﬄ}

r copies

 V V

|ﬄﬄﬄﬄﬄﬄﬄﬄ{zﬄﬄﬄﬄﬄﬄﬄﬄ}

s copies

! R :

ðw

1

; ...; w

r

; v

; ...; v

Þ7!w

1

w

r

...i

...j

v

ð4:11Þ

4.1 Tensor Algebras 99

of the indicated product vector space into the scalar ﬁeld R: Because of the

universality of the canonical mapping / of this product vector space onto V

r;s

considered for two factors after (4.1), one has the following isomorphism:

r; s

is canonically isomorphic to the vector space of all ðr þ sÞ-linear mappings

of V



V



V V ððr þ sÞfactorsÞ into R:

There are simple variants of that proposition. For instance, there is a canonical

isomorphism between V

1;s

and the s-linear mappings of VV (s factors) into

V: A symmetric (see below) tensor g 2 V

0;2

with the property gðv; vÞ0 and

gðv; vÞ¼0 iff v ¼ 0 deﬁnes a scalar product in V and hence converts a general

vector space V into a Euclidean space. Of course, all these mappings are mappings

of vector spaces and do not depend on the actually chosen base in V: In this sense,

scalars, vectors and tensors are called invariant and contra- and covariant,

respectively, entities.

For s ¼ r; one may also consider (4.11) as a bilinear mapping V

0;r

 V

r; 0

! R:

It is easily seen that every bilinear mapping of these spaces into R has the form

(4.11), hence the two spaces are dual to each other:

0;r

¼ðV

r;0



: ð4:12Þ

For r ¼ 1; this is just the duality of V and V



; and, if V is a Euclidean space so that



is identiﬁed with V; then the mapping is the scalar product.

Next, consider mappings of V into another vector space V

: By duality, a

homomorphism H from V to V

induces a homomorphism H



from V



to V



; Hvi¼hH



; vi; v 2 V; w



2 V



: (Recall that a ﬁnite-dimensional vector

space is reﬂexive. Given bases in V and V

; H is represented by a matrix, and H



by its transposed.) If V and V

are isomorphic, so are V



and V



; and there is also

an isomorphic mapping from V



onto V



which is H



1

: Let

H : TðVÞ!TðV

be an isomorphism which in case of decomposable tensors acts like H on each

vector factor and like H



1

on each covector factor. It is easily seen that

commutes with contractions, if it acts on V

0;0

¼ R as the identity mapping

(exercise). The following statement is now rather obvious:

There is a canonical one–one mapping between isomorphisms from V to V

and

isomorphisms from TðVÞ to TðV

Þ which preserve the degree and commute with

tensor contraction. In particular, the automorphism group of V is isomorphic to

the automorphism group of TðVÞ:

The automorphisms of a vector space are also called (regular) transforma-

tions. By the canonical isomorphism between V

1;1

and the space of linear map-

pings (endomorphisms) of V into V (see top of this page) there is a one–one

correspondence of tensors a of degree (1,1) with components given by regular

matrices and automorphisms A of V: These tensors are called transformation

tensors. Sometimes these transformations, which transform a given vector v 2 V in

general in a different one v

¼ Av; in components related to a fixed base v

¼ a

;

are called ‘active coordinate transformations’ while ordinary coordinate

100 4 Tensor Fields

transformations (4.6), which leave all vectors on place and only switch to another

base are called ‘passive coordinate transformations’.

Accordingly, if B : V ! V is any endomorphism of V; it corresponds to a

tensor b 2 V

1;1

whose components do not necessarily form a regular matrix. By an

automorphism A; B is transformed into ABA

1

and the corresponding transfor-

mation of b in TðVÞ is in components b

! a

ða

1

: Endomorphisms of vector

spaces form a ring: if B; B

are two endomorphisms, then B þ B

and BB

are again

endomorphisms. With the Lie product ½B; B

¼BB

 B

B they form also a Lie

algebra. An endomorphism D of TðVÞ is called a derivation,if

1: D preserves the degree: DV

r;s

 V

r; s

;

2: Dðt t

Þ¼ðDtÞt

þ t ðDt

Þ;

3: DC

p;q

¼ C

p;q

D for every tensor contraction:

ð4:13Þ

It is directly seen that, if D; D

are two derivations of TðVÞ ; then ½D; D

 is again a

derivation. The derivations of TðVÞ form another Lie algebra.

The Lie algebra of derivations of TðVÞ is isomorphic to the Lie algebra of

endomorphisms of V; the isomorphism is provided by the restriction B of deri-

vations D to V  TðVÞ:

Proof As an endomorphism, Dkt ¼ kDt; k 2 R: However, kt ¼ k  t in TðVÞ:

From property 2 of (4.13) it follows that Dk ¼ 0 for every k 2 R: Hence, with

property 3, 0 ¼ Dhw



; vi¼hDw



; viþhw



; Dvi for all v 2 V and w



2 V



(exercise). By putting Dv ¼ Bv; it follows Dw



¼B



where B



is the endo-

morphism transposed to B; hw



; Bvi¼hB



; vi: Given B; from these relations D

is determined for decomposable tensors and is uniquely extended by linearity to all

TðVÞ : It is easily seen that B 7!D is a bijection. h

If B

is another endomorphism of V; it is transformed by the derivation D with

Dv ¼ Bv into DB

¼ BB

 B

B ¼½B; B

:

Let P

be the permutation group of the set f1; ...; rg of r numbers and let

P 2P

: Denote by the same letter a mapping P : V

r;0

! V

r;0

: t

...i

7!ðPtÞ

...i

and an analogous mapping P : V

0;r

! V

0;r

: This deﬁnition is obviously

independent of the choice of a base in V: The symmetrization of a tensor of degree

either ðr; 0Þ or ð0; rÞ is

St ¼

P2P

Pt ð4:14Þ

and the alternation is

At ¼

P2P

signðPÞPt; ð4:15Þ

where signðPÞ¼þ1 for an even permutation and signðPÞ¼1 for an odd per-

mutation. A tensor St is called a symmetric tensor and a tensor At is called an

4.1 Tensor Algebras 101

alternating tensor. It is directly seen that these tensors provide symmetric and

alternating multilinear mappings of the product vector space of r factors V



or r

factors V into R:

4.2 Exterior Algebras

Let TðVÞ¼

r¼0

r;0

be the subalgebra of contravariant tensors of TðVÞ and let

IðVÞ be the two-sided ideal of

TðVÞ generated by all elements of the form v 

v; v 2 V; that is, IðVÞ is the linear span of the sets

TðVÞv v  TðVÞ for all

v 2 V: The exterior algebra or Grassmann algebra of V is the graded algebra

KðVÞ¼

TðVÞ = IðVÞ: Then, TðV



Þ is the subalgebra of covariant tensors of TðVÞ

and KðV



Þ¼TðV



Þ=Ið V



Þ: Grassmann was the ﬁrst to introduce the exterior

algebra for the study of subspaces of vector spaces.

KðVÞ is graded in the following way: K

ðVÞ¼R; K

ðVÞ¼V; K

ðVÞ¼

r; 0

ðVÞ; I

ðVÞ¼IðVÞ\V

r;0

for r [ 1 and KðVÞ¼

r¼0

ðVÞ: Since for

every v

; v

2 V the products v

 v

; v

 v

and ðv

þ v

Þðv

þ v

Þ are ele-

ments of I

ðVÞ; it follows that also v

 v

þ v

 v

2 I

ðVÞ : Hence, if the

product in KðVÞ is denoted by ^ ; then v

^ v

¼v

^ v

: It is easily seen that, if

x; r; s 2 KðVÞ and F; G 2 R ¼ K

ðVÞ; then the exterior product ^ in KðVÞ has

all the properties (3.23) of a wedge-product.

Since every decomposable tensor v

v

; r [ 1; containing two con-

secutive equal factors is in IðVÞ and reordering of the factors in TðVÞ leads to

representatives of the same or of the reversed (reversed sign) equivalence class of

TðVÞ = IðVÞ¼KðVÞ; the elements of the exterior algebra KðVÞ may be represented

as linear combinations of

1; e

^^e

¼ e

e

; i

\ \i

; r [0; ð4:16Þ

for any given base fe

g of V: (The ‘base vector’ 1 spans K

ðVÞ¼R; and

1 ^ e

¼ 1e

 e

1 ¼ 0:) Hence, (4.16) forms a base of KðVÞ: It immediately

follows that, if dim V ¼ n; then

dim K

ðVÞ¼



r!ðn  rÞ!

; K

ðVÞR; K

ðVÞ¼f0g for r [ n

ð4:17Þ

and hence dim KðVÞ¼2

: As opposed to the tensor algebra TðVÞ; the exterior

algebra KðVÞ is ﬁnite-dimensional.

Again the canonical mapping /

: V V ! K

ðVÞ : ðv

; ...; v

Þ7!v

^v

has the universality property that, if W is any vector space and w :

V V ! W is any alternating r-linear mapping, then there is a unique linear

mapping w

: K

ðVÞ!W so that w ¼ w

 /

: Up to isomorphisms, /

is uniquely

determined by this universality property.

102 4 Tensor Fields

From the base (4.16) and the properties (3.23) it is seen that the elements of the

exterior algebra KðVÞ may be represented by the linear combinations of alternating

tensors

; t

2 V

r;0

: Likewise, the elements of KðV



Þ may be represented by

the linear combinations

; t

2 V

0;r

: If t 2 K

ðVÞ; r [ 0; then it may be

represented as

t ¼

;...;i

...i

e

¼ r!

\...\i

...i

^^e

: ð4:18Þ

In the middle expressions, the i-sums run independently from 1 to n; but since t

...i

is alternating, only items with distinct i are non-zero, and their r! permutations

appearing in the sums may be summed up into the right expressions.

Consider now e

^^e

; which is a special homogeneous tensor. According

to (4.18), its components consist only of alternating sign factors:

^^e

P2P

signðPÞe

e

As an alternating r-linear mapping from V



V



(r factors) into R like in

(4.11) it yields (cf. the text after (3.23))

^^e

ðw

1

; ...; w

r

Þ¼

detðhw

i

; e

iÞ: ð4:19Þ

By r-linearity and by the expansion rule for determinants, the base vectors in this

relation may be replaced by any set of r linearly independent vectors v

; ...; v

the vector space V ¼ K

: Using tensor contractions, one may also consider the

bilinear mapping K

ðV



ÞK

ðVÞ!R; for which again the right hand side of

(4.19) would follow. However, in order to avoid nasty factorial prefactors in the

exterior calculus, one redefines the prefactor of the latter mapping as

1

^^w

r

; v

^^v

i¼detðhw

i

; v

iÞ: ð4:20Þ

By linearity this mapping (4.20) can be extended to all K

ðV



ÞK

ðVÞ; and this

form comprises all linear functions from K

ðVÞ into R: Hence, K

ðV



Þ is iso-

morphic to the dual vector space to K

ðVÞ :

ðK

ðVÞÞ



 K

ðV



Þ and hence ðKðVÞÞ



 KðV



Þ: ð4:21Þ

Since the dual to a finite direct sum of vector spaces is isomorphic to the direct

sum of their duals, the second relation follows. Note that this duality relation h; i

differs from that transferred from (4.12) via the quotient algebra formation by an

additional prefactor r!: For r ¼ 1 they are equal. In practice, in the exterior cal-

culus only the form (4.20) is used and no confusion can arise.

Consider again dual bases fe

g in V and ff

g in V



; hf

; e

i¼d

: Equation

4.20 yields

^^f

; e

^^e

i¼

P2P

signðPÞd

d

¼ d

...i

...j

: ð4:22Þ

4.2 Exterior Algebras 103

This is called a generalized Kronecker symbol. It is zero, if the sets of integers

g and fj

g are not the same; it is equal to 1, if they are the same and the ordered

set ðj

; ...; j

Þ is an even permutation of the ordered set ði

; ...; i

Þ; and equal to

1; if it is an odd permutation. Let now t 2 K

ðVÞ be any alternating tensor (4.18)

or t

2 K

ðV



Þ; then

...i

^^f

; ti; t

...j

; e

^^e

i: ð4:23Þ

These are the general rules of calculating vector components by projection on the

dual basis, applied to the cases of K

ðVÞ and K

ðV



Þ: For later use, an important

consequence of (4.22) for the relation between tensor algebra and exterior

algebra is

hx; X

^^X

i¼C

1;1

C

r;r

ðx  X

X

Þ; ð4:24Þ

where x is an arbitrary alternating tensor of type ð0; rÞ and X

are vectors.

Next, important endomorphisms of KðVÞ are considered. In (4.17) it was

already noted that K

ðVÞ¼f0g; if r [ dim V: In addition, by deﬁnition,

ðVÞ¼f0g for r\0; KðVÞ¼

1\r\1

ðVÞ: ð4:25Þ

Obviously, the last relation is the same as that given at the beginning of this

section. An endomorphism L of a graded algebra K is called an endomorphism of

degree s; if

L : K

! K

rþs

; 1\r; s\1: ð4:26Þ

For instance, for any u 2 K

ðVÞ; the wedge-multiplication L

: KðVÞ!KðVÞ :

t 7!u ^t is an endomorphism of degree s: An endomorphism is called a deriva-

tion,if

Dðt ^ t

Þ¼ðDtÞ^t

þ t ^ðDt

Þ for all t; t

2 K; ð4:27Þ

it is called an anti-derivation,if

Dðt ^ t

Þ¼ðDtÞ^t

þð1Þ

t ^ðDt

Þ for all t 2 K

; t

2 K: ð4:28Þ

By repeated application of (4.28) and realization of the associativity of the algebra

KðVÞ one gets for an antiderivation of a decomposable element

Dðv

^^v

Þ¼

i¼1

ð1Þ

iþ1

^^ðDv

Þ^^v

: ð4:29Þ

104 4 Tensor Fields

As opposed to the derivatives in analysis considered in Sect. 2.3, the derivations of

an algebra (or a ring) are always endomorphisms. Examples of derivations of

degree 0 are the linear derivation of the algebra F

of germs of real functions,

(3.10), (formally, any algebra can be considered as a graded algebra, where all

subspaces of non-zero degree are f0g), the derivation of the algebra CðMÞ of

smooth real functions, (3.15), by any tangent vector ﬁeld and the derivation (4.13)

of a tensor algebra. An example of an anti-derivation of degree 1 is the exterior

differentiation d of r-forms, (3.25).

With regard to mutually dual algebras Kð VÞ and ðKðVÞÞ



¼ KðV



Þ; the trans-

posed of the above mentioned endomorphism L

; u 2 KðVÞ may be considered. It

is denoted by i

: KðV



Þ!KðV



Þ : hi



; t

i¼ht



; L

i¼ht



; u ^ t

i for every t

2 KðVÞ:

ð4:30Þ

It is called the interior multiplication by u in KðV



Þ: (For t



; t

2 K

ðVÞ¼R it

follows from the remark in parentheses after (4.16) that i



¼ 0:) An example is

the Hodge operator considered in Sect. 5.1.

For v 2 V; the endomorphism i

is an anti-derivation of degree 1 on KðV



Þ:

Proof Since L

is of degree 1 according to its description after (4.26), it follows

from (4.30) that i

is of degree 1: Consider decomposable elements. From the

deﬁnition of i

; hi

ðw

1

^^w

r

Þ; v

^^v

i¼hw

1

^^w

r

; v

^ v

^v

i¼detðhw

i

; v

iÞ: If one replaces in (4.29) D with i

and v

with w

i

and

inserts the right hand side of the obtained relation into hi

ðw

1

^^w

r

Þ;

^^v

i; one obtains the expansion of the same determinant with respect to

its ﬁrst line. h

If w : V ! V

is a homomorphism of vector spaces, it extends to a homo-

morphism (push forward) w



: KðVÞ!KðV

Þ of algebras as w



ðv

^^v

Þ¼

wðv

Þ^^wðv

Þ and further by linear extension. It also yields by duality a

homomorphism (pull back) w



: KðV



Þ!KðV



Þ via hw



ðu



Þ; ti¼hu



; w



ðtÞi

for all u



2 KðV



Þ and all t 2 KðVÞ:

The section is closed with three simple useful theorems which are easily proved

by completion of the considered sets of vectors to a base of the vector space and by

observing that for any base fv

g of an n-dimensional vector space V the wedge-

product v

^^v

is non-zero, and that an expansion of a vector into a base has

unique components.

A set of vectors v

; ...; v

2 V is linearly dependent, iff v

^^v

¼ 0:

Let fv

g and fv

g be two sets each of r vectors of V so that

i¼1

^ v

¼ 0:

If the v

are linearly independent, then the v

may be linearly expanded into the v

;

i¼1

; with a symmetric coefﬁcient matrix, w

¼ w

Let fv

g be a set of r linearly independent vectors of V and let t 2 K

ðVÞ: Then;

t  0 mod ðv

; ...; v

Þ; that is t ¼ v

^ u

þþv

^ u

with certain u

s1

ðVÞ ; iff v

^^v

^ t ¼ 0:

4.2 Exterior Algebras 105

Prove the theorems as an exercise. For s ¼ 2 the last one was considered in the

Frobenius theorem for Pfafﬁan systems, Sect. 3.6.

4.3 Tensor Fields and Exterior Forms

In Sect. 3.4 tangent vector ﬁelds on a manifold M were introduced by assigning a

tangent vector to every point x 2 M: While smoothness of real functions on M

could naturally be deﬁned with the help of coordinate neighborhoods on the basis

of a well-deﬁned atlas structure for M (pseudo-group of transition functions), this

is not so simple for a general vector ﬁeld. Although an n dimensional vector ﬁeld

may be given by n real component functions, these components depend on the

choice of a base in the vector space at each point of M; and for the concept of

smoothness some rules are needed how the bases of vector spaces on neighboring

points x of M should be related. This will be ﬁnally worked out in Chap. 7 with the

concept of ﬁber bundles. For the special case of tangent space the problem was

solved by relating the bases of the tangent spaces to the coordinates on coordinate

neighborhoods of M via considering the action of a tangent vector on real func-

tions on M: Then, the bases of cotangent spaces were related to those of tangent

spaces via considering the action of cotangent vectors (1-forms) on tangent vec-

tors. Since the base of a tensor algebra or an exterior algebra on a vector space is

determined by the base of the vector space, tensor ﬁelds and exterior ﬁelds on

tangent and cotangent spaces, which are sections of corresponding ﬁber bundles

considered in Chap. 7, can be treated here without the concept of ﬁber bundles.

Let M be a manifold, T

the tangent space and T



the cotangent space at point

x 2 M: Consider the sets

r;s

ðMÞ¼

[

x2M

ðT

r;s

: tensor bundle of type ðr; sÞ over M; ð4:31Þ



ðMÞ¼

[

x2M

ðT



Þ : exterior r bundle over M: ð4:32Þ

In Chap. 7 a topology will be introduced into these sets to provide them with the

special manifold structure of bundles.

Consider sets of r 1-forms x

; i ¼ 1; ...; r and s tangent vector ﬁelds X

;

j ¼ 1; ...; s on M; and multilinear mappings t : T



ðMÞT



ðMÞTðMÞ

TðMÞ!CðMÞ with r factors T



ðMÞ and s factors TðMÞ: According to

(4.11), at each point x 2 M the mapping is given by ðx

; ...; x

; n

; ...; n

Þ7!

...i

...j

ðxÞx

x

n

2 R: Recall that in coordinate neighborhoods in M

the base forms dx

in T



ðMÞ and the base vectors o=ox

in TðMÞ are smooth.

Hence, the mapping is into CðMÞ; if the component functions t

...i

...j

ðxÞ are smooth

functions of x: In this case,

106 4 Tensor Fields