Eschrig H. Topology and Geometry for Physics

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Proof Apply r

to tðX

; ...; X

Þ¼C

1;1

C

s;s

ðt  X

X

Þ and observe

(4.13) for D ¼r

: h

One now may apply r a second time and obtain ðr

tÞðX

; ...; X

; X; YÞ¼

ðr

ðrtÞÞð X

; ...; X

; XÞ; or recursively more generally

ðr

tÞð...;X

;...;X

Þ¼ðr

ðr

n1

tÞÞð...;X

;...;X

n1

Þ: ð7:28Þ

Like in the general case of sections in a vector bundle, the tensor ﬁeld t is a

parallel tensor ﬁeld,ifr

t ¼ 0 for all X 2 T

ðMÞ at all x 2 M; that is, rt ¼ 0:

The alert reader might be intrigued by the question why there are two structure equations in

the case of a linear connection on M while there is in general only one (the second). Some insight

into this situation is obtained by considering generalized afﬁne connections as introduced by

Kobayashi and Nomizu. These are connections on the afﬁne frame bundle considered in Sect. 7.2.

Take a connection form

x deﬁning a connection

C on the afﬁne frame bundle AðMÞ: It is a

pseudo-tensorial 1-form of type ðAd; aðm; RÞÞ: Pull it back to the linear frame bundle LðMÞ by

the homomorphism c considered in Sect. 7.2. According to the semi-direct sum aðm; RÞ¼

glðm; RÞR

one obtains



xÞ¼x þ u;

where x is a pseudo-tensorial 1-form of type ðAd; glðm; RÞÞ and u is of type ðGlðm; RÞ; R

Þ: It

acts linearly on R

(on the last column of the ðm þ 1Þ ðm þ1Þ-matrix representation given in

Sect. 7.2) and produces R

-vectors, hence it can be represented by an R

-tensor t

of type ð1; 1Þ:

On LðMÞ; the vertical spaces are isomorphic to glðm; RÞ which does not have the m þ1st column,

hence u is horizontal on LðMÞ and constitutes a tensorial 1-form of type ðGlðm; RÞ; R

Þ there.

As a pseudo-tensorial 1-form of type ðAd; glðm; RÞÞ; x deﬁnes a linear connection C on LðMÞ:

The mapping between connections,

C 7!ðC; t

Þ; where t

is any tensor ﬁeld of type ð1; 1Þ on M

turns out to be one–one, it comprises a pushed forward homomorphism b



C 7!C (from

b : Aðm; RÞ!GLðm; RÞ). Take the exterior derivative of the above displayed relation (it

commutes with the homomorphism c



; see (4.43)) and obtain c



ðd

xÞ¼dx þ du: Let X; Y be

two horizontal vector ﬁelds on LðMÞ; then the right hand side of the last equation

yields hðdx þduÞ; X ^ Yi¼hðX þDuÞ; X ^ Yi: Its left hand side yields, with the structure

equation of

C; hd

x; c



ðXÞ^c



ðYÞi ¼ ½h

x; c



ðXÞi; h

x; c



ðYÞi þh

X; c



ðXÞ^c



ðYÞi: Since X; Y

are horizontal for C; hx; Xi¼hx; Yi¼0 and h

x; Xi¼hu; Xi; h

x; Yi¼hu; Yi: However,

is Abelian and hence ½hu; Xi; hu; Yi ¼ 0 and hc



ðd

xÞ; X ^Yi¼hd

x; c



ðXÞ^c



ðYÞi ¼

X; c



ðXÞ^c



ðYÞi ¼ hc



XÞ; X ^Yi: In total,



XÞ¼X þ Du:

Use again the structure equation of

C on AðMÞ; d

x ¼

x ^

x þ

X; pull it back to LðMÞ and

insert x þ u for c



xÞ: Split the resulting equation dðx þ uÞ¼x ^ x  x ^ u þ X þ Du into

the glðm; RÞ-components and the R

-components and obtain ﬁnally

du ¼x ^u þ Du; dx ¼x ^ x þ X:

In view of this result, a generalized afﬁne connection

C on M is called an afﬁne connection, if the

-valued 1-form u is the canonical form h on LðMÞ: In this case the above relations are just the

structure equations of a linear connection C on M: The canonical form h as introduced by (7.5)

maps the horizontal space identical into the horizontal space, hence the corresponding tensor t

7.7 Linear and Afﬁne Connections 237

the unit tensor, and one is left with a one–one correspondence between afﬁne connections

C and

linear connections C on M: Therefore these two names are used synonymously in the literature.

However, if one uses the principal ﬁber bundle AðMÞ instead of LðMÞ to deﬁne a linear con-

nection, the two structure equations are again merged into a single one like in the general case.

7.8 Curvature and Torsion Tensors

Let a linear connection C be given on a manifold M; and let X; Y be two tangent

vector ﬁelds on M: Let X



; Y



be lifts of X; Y into LðMÞ; and consider hH; X



^ Y



with the torsion form H of the connection C: Since H is a tensorial 2-form of type

ðGlðm; RÞ; R

Þ; this expression deﬁnes a vector T

2 R

at every point p 2 LðMÞ:

With the linear mapping uðpÞ introduced in Sect. 7.2, T

is mapped to a tangent

vector of T

pðpÞ

ðMÞ; and the whole result depends linearly on X ^ Y: Hence it may

be expressed by a tensor ﬁeld T of type (1,2) on M as

hT; X ^ Yi¼uhH; X



^ Y



i; ð7:29Þ

which is to be understood that the value of the left hand side at x ¼ pðpÞ2M is

given by uðpÞ applied to the value at p of the argument of u on the right hand side.

It is easily seen, that the result at x does not depend on the actual point p 2 p

1

ðxÞ:

Indeed, let p

¼ pg; g 2 Glðm; RÞ: Then as lifts, X



¼ðR



; Y



¼ðR



The right hand side of (7.29)atp

is uðp

ÞhH

; X



^ Y



i¼uðp

ÞhH

; ðR



^ðR



i¼uðp

ÞhR



ðH

Þ; X



^ Y



i¼uðp

Þhg

1

; X



^ Y



i¼uðp

Þg

1

; X



^ Y



In the last but one equality it was used that H is a tensorial 2-form of type

ðGlðm; RÞ; R

Þ (compare (7.7) with Glðm; RÞ instead of Ad). Now, since

uðp

Þg

1

¼ uðp

1

Þ¼uðpÞ; the result is the same as that at p: Hence, for every

x 2 M; (7.29) is uniquely deﬁned by the right hand side and is a tangent vector of

ðMÞ for every pair of tangent vectors X; Y; which means that T 2T

1;2

ðMÞ is a

tensor ﬁeld of type ð1; 2Þ alternating in the lower indices (in coordinate repre-

sentation). It is called the torsion tensor ﬁeld or simply also the torsion of the

linear connection C on M: Equation 7.29 is called the torsion operation on the

pair X; Y:

In an analogous manner the curvature operation on a pair X; Y is deﬁned.

Since the curvature form X is a tensorial 2-form of type ðAd; glðm; RÞÞ; the cur-

vature operation at x is an element of the Lie algebra glðm; RÞ and hence a (not

necessarily regular) linear transformation of tangent vectors. This transformation

of a tangent vector ﬁeld Z is deﬁned as

CðhR; X ^ YiZÞ¼uðhX; X



^ Y



iðu

1

ZÞÞ; ð7:30Þ

where C means the contraction (4.9) of the tensor product (in local coordinates of

the lower index of the tensor hR; X ^ Yi of type ð1; 1Þ with the upper index of Z).

On the right hand side, hX; X



^ Y



i2glðm; RÞ and u

1

Z 2 R

: Hence, the

argument of u is again in R

which is mapped by u into TðMÞ: The independence

238 7 Bundles and Connections

of the right hand side on p 2 p

1

ðxÞ to which X and Y are lifted is seen in the same

manner as above, only now R



ðX

Þ¼Adðg

1

ÞX

¼ g

1

g and uðp

1

ðuðpÞg

1

¼ guðpÞ

1

: The result depends multi-linearly on X; Y; Z and is a

tangent vector ﬁeld on M: Hence, it may be represented by a tensor ﬁeld of type

ð1; 3Þ which is alternating in its ﬁrst two lower indices in a coordinate represen-

tation. It is called the curvature tensor ﬁeld or simply also the curvature of the

linear connection C on M:

Of course, a neighborhood of x sufﬁces to deﬁne the curvature and torsion

tensors at x:

The torsion and curvature operations can be expressed in terms of covariant

derivatives as

hT; X ^ Yi¼r

Y r

X ½X; Y; hR; X ^ Yi¼½r

; r

r

½X;Y

: ð7:31Þ

Proof Start with hT

; X

^ Y

i¼uðpÞhH

; X



^ Y



i; pðpÞ¼x: With (7.21), since



and Y



are horizontal lifts and hence hx; X



i¼0 ¼hx; Y



i; hH

; X



^ Y



i¼

hdh

; X



^ Y



i¼L



hh; Y



iL



hh; X



ihh

; ½X



; Y





i¼X



hh; Y



iY



hh; X



i

; ½X



; Y





i: With (7.5), the last term yields uðpÞhh

; ½X



; Y





i¼p



ð½X



; Y





Þ¼

½X; Y

: It remains to consider uðpÞðX



hh; Y



iÞ: Take ﬁrst hh

; Y



i¼uðpÞ

1

ðY

which is a vector in R

whose components are functions of p: Let /

ðxÞ; /

ðxÞ¼x;

be a curve in M through x to which X

is tangent, and let /



ðpÞ be its lift through p

which is a curve in the frame bundle LðMÞ to which X



is tangent. Hence,



hh; Y



i¼lim

d!0

uð/



ðpÞÞ

1

ðY

ðxÞ

ÞuðpÞ

1

ðY

and

uðpÞðX



hh; Y



iÞ ¼ lim

d!0

uðpÞuð/



ðpÞÞ

1

ðY

ðxÞ

ÞY

Here, uð/



ðpÞÞ

1

maps Y

ðxÞ

into R

and uðpÞ maps this image into T

ðMÞ: Since



ðpÞ and p ¼ /



ðpÞ are connected by a horizontal path in LðMÞ; the two map-

pings realize a horizontal transport of Y

ðxÞ

from /

ðxÞ to x ¼ /

ðxÞ: Call this

transport U



ðt;0Þ

and compare to (7.17) to see that the result is ðr

YÞ

: Putting

together these ﬁndings proves the ﬁrst relation (7.31).

Now, start with CðhR

; X

^ Y

iZ

Þ¼uðpÞðhX

; X



^ Y



iðuðpÞ

1

ÞÞ:

Since X



and Y



are again horizontal lifts, hX; X



^ Y



i¼hdx; X



^ Y



i¼



hx; Y



iL



hx; X



ihx; ½X



; Y



i ¼ hx;

½X



; Y



i: (Recall that x anni-

hilates horizontal vectors.) Let A



be the fundamental vector ﬁeld on LðMÞ

which at p equals A



½X



; Y





; so that A ¼hx

; A



i is an element

of glðm; RÞ: So far, CðhR

; X

^ Y

iZ

Þ¼uðpÞðhx;

½X



; Y





ihh

; Z



iÞ:

7.8 Curvature and Torsion Tensors 239

F ¼hh; Z



i is an R

-valued function on LðMÞ which is tensorial of type

ðGlðm; RÞ; R

Þ: Hence,



F ¼ lim

t!0

Fðp expðtAÞÞ FðpÞ

¼ lim

t!0

expðtAÞFðpÞFðpÞ

¼AFðpÞ:

Therefore,

hx

;

½X



; Y





ihh

; Z



iÞ ¼

½X



; Y





hh; Z



i¼ð½X



; Y







½X



; Y





hh; Z



i: From the ﬁrst part of the proof above, uðpÞð

½X



; Y





hh; Z



iÞ ¼ ðr

½X;Y

ZÞ

for the horizontal vector

½X



; Y





: On the other hand, again with the ﬁrst part of

the proof and using the horizontality of X



; Y



; one obtains uðpÞðX



ðY



hðZ



ÞÞÞ ¼

uðpÞðX



ðuðpÞ

1

 uðpÞðY



hh; Z



iÞÞÞ ¼ uðpÞðX



ðuðpÞ

1

ðr

ZÞ

ÞÞ ¼ uðpÞðX



; ðr

ZÞ



iÞ ¼

ðr

ZÞ

: Putting everything together and observing that one may formally write

ðr

ZÞ

¼ Cðr

 ZÞÞ

acts like a transformation tensor of type ð1; 1Þ), the

proof of the second relation (7.31) is completed, since Z was chosen completely

arbitrarily. h

7.9 Expressions in Local Coordinates on M

In this section, ﬁnally local coordinate expressions are derived for the forms, the

covariant derivative and the torsion and curvature tensors of a linear connection on

M: For the sake of simplicity of notation, the same letter is used for points in

manifolds and in corresponding coordinate spaces. As in Sect. 7.2, local coordi-

nates p ¼ðx

; X

; i ¼ 1; ...; mÞ; so that x ¼ðx

; ...; x

Þ and X

ðo=ox

Þ;

detðX

Þ 6¼ 0; are introduced in p

1

ðUÞLðMÞ where U is a coordinate neigh-

borhood of x in M: The inverse of the ðm mÞ-matrix ðX

Þ is denoted by ð

Þ;

so that

¼ d

: ð7:32Þ

In addition, in R

the natural base fe

; ...; e

g is introduced, for which

h ¼

: ð7:33Þ

For any p ¼ðx

; X

Þ2U; the mapping uðpÞ : R

! T

ðMÞ; x ¼ pðpÞ; maps e

ðo=ox

(see Sect. 7.2). Let



ðpÞ



ðpÞ

be any tangent vector in T

ðLðMÞÞ: Its projection to the tangent space on M;

ðMÞ; is p



ðY



Þ¼

ðpÞðo=ox

; hence

240 7 Bundles and Connections

; Y



i¼uðpÞ

1

ðp



ðY



ÞÞ ¼

ðpÞY

ðpÞÞe

;

since uðpÞ

1

ðo=ox

ðpÞe

: Comparison with the representation (7.33)

yields h h

; Y



i¼

ðpÞY

ðpÞ or, with hdx

; o=ox

i¼d

;

ðpÞdx

or in short h

ð7:34Þ

as the local coordinate expression of the canonical form h on LðMÞ: As an

-valued 1-form, the local coordinate expression

of h has an upper index i as

an R

-vector and a lower index j as a 1-form according to the general local

coordinate representation (3.24) of an exterior form.

Consider the transition properties of h between two overlapping coordinate

neighborhoods U

\ U

3 x: According to (3.14), the tangent vector X

on M

transforms as X

ðw

where w

is the Jacobian matrix of the coor-

dinate transformation given by (3.6). Hence,

transforms like a cotangent vector,

ðw

1

; in order that

¼ d

: With the second

relation (3.11) this ensures that

: As seen, h

behaves

indeed like a tensor of type ð0; 1Þ on M; which yields another justiﬁcation to call it

a tensorial 1-form. (Recall that originally h was introduced in Sect. 7.3 as a 1-form

on LðMÞ:)

The connection form x of a linear connection C on M is a glðm; RÞ-valued

1-form. For its corresponding local coordinate expression the natural base fE

g in

glðm; RÞ is needed, which consists of matrices with a unity in the ith column and

jth row and zeros otherwise (p. 219). The analogue of (7.33)is

x ¼

: ð7:35Þ

On a coordinate neighborhood U

 M; the analogue of (7.34) is ﬁrst considered

for the local connection forms x

only, which are pull-backs from the canonical

local section s

 LðMÞ to U

ðxÞdx

or in short x

The components C

of the local connection form are called Christoffel symbols.

Unlike the components of the canonical form h; the Christoffel symbols to

not form a tensor on M: The local connection forms x

must have the transi-

tion properties (7.4), that is, the pairing with a tangent vector ﬁeld X on M;

ðw

; must obey the relation

7.9 Expressions in Local Coordinates on M 241

; X

i¼

lmn

ðw

1

; X

iðw

ðw

þhð#

; X

iðw

where for X

the local connection forms x

and x

were the pulled back con-

nection forms from p

¼ s

ðxÞ and p

¼ s

ðxÞ given by the canonical local sec-

tions. (Recall also that w

¼ w

1

In order to ﬁnd the coordinate transition expression #

¼ w



ð#Þ of the

Maurer–Cartan 1-form # of Glðm; RÞ; the way (7.4) was obtained has to be

reconsidered. In the coordinate neighborhood U

3 x; let s

be the canonical

local section which maps x to the frame ðx; ðo=ox

Þ; ...; ðo=ox

ÞÞ: Clearly,

ðx; ðo=ox

Þ; ...; ðo=ox

ÞÞ ¼ ðx;

ðw

1

ðo=ox

Þ; ...; ðw

1

ðo=ox

Þ¼ðx; ðo=ox

Þ; ...;

ðo=ox

ÞÞw

as was used in (7.4). Any frame p of p

1

ðxÞ is obtained by acting

from the right on s

ðxÞ with an element g of the Lie group Glðm; RÞ; p ¼ s

ðxÞg:

(Recall that according to property 2 on p. 215 the same group acts on x as an

element of glðm; RÞ by the adjoint representation.) Use again the natural base E

glðm; RÞ as above. An element g in natural coordinates of the Lie group Glðm; RÞ

is represented by g

; detðg

Þ 6¼ 0: Let ð

Þ be the matrix inverse to ðg

Þ: The

Maurer–Cartan form maps left invariant tangent vector ﬁelds on Glðm; RÞ into

glðm; RÞ and maps ð o = o g

2 T

ðGlðm; RÞÞ to E

: h#

; ðo=og

i¼E

: Let G

the left invariant tangent vector ﬁeld which at e is G

¼ðo = og

; that is, G

ðo=og

: It must also hold that h#

; G

i¼E

: As a glðm; RÞ-valued

1-form, write # ¼

klmn

and use hdg

; ðo=og

Þi ¼ d

: There must

klmn

gkn

¼ E

which ﬁnally results in #

gkn

¼ g

or #

klmn

: Now,

¼ w



ð#Þ¼#

klmn

ðw

1

dðw

; ð7:36Þ

where

dðw

ðdw

: ð7:37Þ

Putting everything together results in

bjk

lmn

amn

ðw

1

ðw

1

ðw

ðdw

1

; ð7:38Þ

which shows that C

is indeed not a tensor on M (whence x was called a pseudo-

tensorial 1-form).

In local coordinates p ¼ðx

; X

Þ; a vertical tangent vector on LðMÞ has the form



ðo=oX

Þ; and since hdx

; ðo=oX

Þi ¼ 0; the pairings of local connec-

tion forms with vertical tangent vectors vanish. The connection form x itself must,

242 7 Bundles and Connections

however, have the properties 1 and 2 given on p. 215. Hence, x must consist of a

term which, if paired with vertical tangent vectors, restores these properties 1 and

2 and of another term which pulls back to x

: It is easily found that the right

expression is

x ¼

ijk

: ð7:39Þ

Indeed, consider the fundamental vector ﬁeld ðX



¼ R



ðE

Þ which on the

canonical local section s

with local coordinates s

ðxÞ¼ðx

; d

Þ is ðX



ðxÞ

ðo=oX

ðxÞ

: A general point p on the ﬁber over x is p ¼ðx

; X

Þ¼s

ðxÞg; g ¼ X

in local coordinates. A fundamental vector ﬁeld is a left invariant vector ﬁeld on

Glðm; RÞ; hence ðX



¼ gðX



ðxÞ

ðo=oX

: Now,

hx; ðX



i¼

klmrs

hdX

; ðo=oX

ÞiX

klm

¼ E

and prop-

erty 1 is fulﬁlled. Property 2 is directly read off the factor at E

in (7.39), since the

second term of (7.38) vanishes for a ¼ b: Moreover, since s



pulls back from X

; it pulls the ﬁrst term of (7.39) back to zero and the second term to the

expression for x

introduced after (7.35). Hence, (7.39) is the ﬁnal local coor-

dinate expression for the linear connection form.

Every transformation step from (7.4)to(7.38) was one–one. Hence, symbols C

transforming according to (7.38) yield local connection forms x

which obey (7.4).

On the other hand, as it was seen there, local connection forms x

obeying (7.4)

deﬁne uniquely a connection form x on LðMÞand hence a linear connection C on M:

Symbols C

in local coordinates having the transition properties (7.38) deﬁne

uniquely a connection form through (7.39) and thus a linear connection C on M:

Next, let X

¼ðo=ox

Þ be a tangent vector on U

and let X



¼ðo = ox

ðX



ðo=oX

be its horizontal lift through p ¼ðx

; X

Þ: Then, 0 ¼

; X



i¼

ðX



: Multiplication with X

and summation

over i yields ðX



¼

and hence





kmn



: ð7:40Þ

This expression only now determines the splitting of a general tangent vector on

LðMÞ into its horizontal and vertical parts, ﬁrst mentioned on p. 219, by giving the

structure of horizontal vectors in terms of local coordinates.

Now, the tangent bundle ð TðMÞ; M; p

; R

; Glðm; RÞÞ ¼ TðMÞ associated with

the frame bundle LðMÞ is considered where Glðm; RÞ acts on the vector space R

the identical representation, and LðMÞ is provided with a linear connection C: Let

f : LðMÞ!R

be a R

-valued function with the property f ðpgÞ¼g

1

f ðpÞ: It may

be understood to be a tensorial 0-form of type ðGlðm; RÞ; R

Þ: In combination with

7.9 Expressions in Local Coordinates on M 243

the canonical local section s

in LðMÞ it deﬁnes a local section f

: U

! TðMÞ :

x 7!ðs

ðxÞg; g

1

f ðs

ðxÞÞÞ on U

in the tangent bundle TðMÞ; where the section point

for x corresponds tothe tangent vector with components f

ðs

ðxÞÞin the frame d

given

by the natural base fe

; ...; e

g at x: In particular, the functions f

: p ¼ðx

; X

Þ7!

for given i; that is, f

by matrix multiplication to the frame coordinates,

provide this property. The section f

consists of the tangent vector ðo=ox

Þ at every

x 2 U

: According to (7.17), its directional derivative along a path with horizontal

tangent vector X is r

ðo=ox

Þ: Apply the horizontal vector X



from (7.40)tof

and

obtain

o=ox

ðo=ox

Þ¼

klmn

ðo=oX

klmnrs

ðd

where in the second equality (2.26) was used. By reinserting ðo=ox

Þ¼

in the last expression, the ﬁnal result

o=ox

ðo=ox

Þ¼

ðo=ox

Þð7:41Þ

is obtained. Replacing in these considerations the tangent bundle TðMÞ by the

cotangent bundle T



ðMÞ replaces f

by f

¼ X

only and results in

o=ox

¼

: ð7:42Þ

This analysis underlines the role of the frame bundle as principal ﬁber bundle with

which tangent, cotangent and general tensor bundles over M are associated. Would

one try to deﬁne connections directly on those bundles, the deﬁnition would

unavoidably depend on the used local coordinate systems in a quite involved way.

The use of the frame bundle makes it possible to deﬁne linear connections inde-

pendently of local coordinate systems, and in addition leads to general forms of

coordinate expressions. (It was invented by E. Cartan.)

The properties (7.18, 7.19)ofcovariant derivatives can now be used to obtain

the local coordinate expressions of the derivatives of general tensor ﬁelds as

sections of tensor bundles, from (7.41, 7.42). With the general local coordinate

expression (4.33) one gets

tðxÞðÞ

...i

...j

¼ X

...i

...j

l¼1

...i

l1

lþ1

...i

...j



l¼1

...i

...j

l1

lþ1

...j

ð7:43Þ

The notation

o=ox

k tðxÞ

...i

...j



¼ðrtðxÞÞ

...i

...j

¼ t

...i

...j

ðxÞð7:44Þ

is generally used. The tensor contraction of the right hand side with any tangent

vector X gives (7.43) back, which shows that the expression in parentheses on the

right hand side of (7.43) forms the components of a tensor of type ðr; s þ1Þ; the

244 7 Bundles and Connections

covariant differential of t: Higher order covariant differentials are recursively

obtained:

ðr

tÞÞ

...i

...j

...k

¼ t

...i

...j

;...;k

: ð7:45Þ

Compare (7.27, 7.28). These are the generalizations of tensor gradients from

trivial connections for which the Christoffel symbols vanish. (They do not vanish

even in ﬂat connections, if general non-linear coordinates are used.)

The local coordinate expressions of the torsion and curvature tensors are now

straightforwardly obtained from (7.31). First write the left hand sides as

hT; X ^ Yi¼T

; hR; X ^ Yi¼R

jkl

; ð7:46Þ

and then use (7.41) and (7.18, 7.19) on the the right hand sides to ﬁnd (exercise)

¼ C

 C

ð7:47Þ

and

jkl



þ C

 C

: ð7:48Þ

For a smooth function F on M one also directly ﬁnds that

;j;k

 F

;k; j

¼ T

ð7:49Þ

and for a tangent vector ﬁeld X that

;l;k

 X

;k;l

¼ R

jkl

 T

; j

: ð7:50Þ

Thus, the covariant derivatives of functions commute, if the torsion of the linear

connection vanishes, and the covariant derivatives of vector ﬁelds commute, if the

linear connection is ﬂat and torsion free.

A smooth curve xðtÞ in M which locally solves the equations

¼ 0; i ¼ 1; ...; m ð7:51Þ

is called a geodesic. The vector X ¼ðdx

=dtÞðo=ox

Þ is tangent to this curve. From

(7.43) it follows that r

X ¼ 0:

The tangent vector to a geodesic is parallel transported to itself on the

geodesic.

Finally, the Lie derivative (4.36) is compared to the covariant derivative (7.17).

A direct comparison of (3.37) with the ﬁrst relation (7.31) yields immediately

Y ¼r

Y r

X hT; X ^ Yið7:52Þ

for any tangent vector ﬁeld Y; while for any 1-form r

7.9 Expressions in Local Coordinates on M 245

ðL

rÞ

¼ðr

rÞ

þ r

 T

ð7:53Þ

is an exercise. While in Fig. 4.1 for the Lie derivative the transport is along the

ﬂow of X (local 1-parameter group) without a twist, in Fig. 7.8 for the covariant

derivative it is horizontally in the direction of X:

Reference

1. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Interscience, vols. I and II.

New York (1963, 1969)

246 7 Bundles and Connections