Eschrig H. Topology and Geometry for Physics

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1

, and consider the corresponding isotropy group at e 2 G. It is given as

fðg

; g

Þjg

1

¼ eg implying g

¼ g

. Hence, the isotropy group ðG  GÞ

the diagonal D ¼fðg

; g

Þjg

2 GgG. Moreover, G G acts transitively on

G, since for arbitrary g; g

2 G there is g

¼ g

1

¼ðg

; gÞg. Hence, ðG 

GÞ=D ! G : ðg

; g

ÞD 7!g

1

is a diffeomorphism of manifolds. By choosing

ðg

1

; g

1

Þ2D for a representative of the coset ðg

; g

ÞD one ﬁnds ðg

1

; eÞ

which together with the coset is mapped to g

1

by the above diffeomorphism, so

that the diffeomorphism is also a homomorphism of groups and hence it is a Lie

group isomorphism, ðG GÞ=D  G. Consequently, if the Lie group G and hence

also D is compact, G ðG  GÞ=D itself can be provided with an invariant metric

and thus be made into a homogeneous Riemannian manifold.

Consider the Lie algebra g  T

ðGÞ of the Lie group G and introduce on it the

symmetric 2-form

jðX; YÞ¼trðadðXÞadðYÞÞ; X; Y 2 g: ð9:18Þ

It is called the Killing form or Killing–Cartan form of g. Since the elements of

the Lie algebra g are left invariant vector ﬁelds on G,(9.18) is clearly an invariant

2-form:

jðX; YÞ¼jðl

g

X; l

g

YÞ for all g 2 G; ð9:19Þ

where l

g

is the left translation of tangent vectors from e to g in G like in Sect. 6.1.

Recall from that section, that after choosing a base fX

; ...; X

g in g, one gets

adðX

ÞX

¼½X

; X

¼

k¼1

with the structure constants c

of the Lie group

G. Thus, adðX

¼ c

is cast into an ðm mÞ-matrix acting on the m-dimensional

vector space g ¼ span

; ...; X

g, the composition of adðYÞ with adðXÞ

becomes the matrix multiplication and the trace becomes the matrix trace. From

the theory of Lie algebras [2] and citations therein it is known that the Killing form

j is non-degenerate, iff the Lie algebra g is semi-simple; it is negative deﬁnite, iff g

is moreover the Lie algebra of a compact Lie group G: Hence,

gðX; YÞ¼jðX; YÞð9:20Þ

is an invariant metric making a compact semi-simple Lie group into a homoge-

neous Riemannian manifold, and it is an invariant indeﬁnite metric making a

non-compact semi-simple Lie group into a generalized homogeneous Riemannian

manifold.

In the above considered simple case G ¼ OðnÞ, the Lie algebra is g  R

n1

, and

after introducing a standard orthonormal base in R

n1

, gðX

; X

Þ¼g

becomes the

unit matrix, related to standard orthogonal local coordinates on the sphere S

n1

In physics, a closed ﬁnite piece of a Riemannian manifold with ﬁxed time-

independent distances between all of its points is called a rigid body. The pecu-

liarity of a homogeneous manifold is that a piece of it as a rigid body can move

through it without deformation. For that reason, after Riemann’s habilitation talk

9.2 Homogeneous Manifolds 307

where he introduced his revolutionary new concept of geometry, many scholars

including Helmholtz thought that physical relevance should be restricted to

homogeneous manifolds, because the free motion of rigid bodies in space was still

a dogma based on Kant’s absolute space and time. At least after Einstein’s theory

of relativity it is known that rigid bodies are an abstraction from the real world,

only possible in the limit of inﬁnite velocity of light. Nevertheless, as this limit is

often a very good approximation, homogeneous manifolds may continue to have

relevance in kinematics and classical physics. Besides, Lagrangian Grassmannians

are important homogeneous manifolds forming subspaces of the phase space of

classical mechanics.

9.3 Riemannian Connection

Further on, M is a generalized Riemannian manifold with the metric form g. Like

any tensor ﬁeld, g can be considered as a section in a tensor bundle over M which

is associated with the frame bundle ðLðMÞ; M; p; Glðm; RÞÞ as its principal ﬁber

bundle. In Sect. 7.7, linear connections on M were introduced as connections on

LðMÞ, and covariant derivatives r

t of tensor ﬁelds t in the direction of the

tangent vector X as well as covariant differentials rt were associated with linear

connections. A linear connection C on M is called a metric connection,ifrg ¼ 0

on M, a metric connection is called a Riemannian connection,orLevi–Civita

connection, if it is torsion free, T ¼ 0 on M. This is subject of the Fundamental

Theorem of Riemannian Geometry:

Every generalized Riemannian manifold allows for exactly one Riemannian

connection. It is deﬁned by the following expression for r

Y valid for every

X; Y; Z 2 g:

2gðr

Y; ZÞ¼XgðY; ZÞþYgðX; ZÞZgðX; YÞ

 gðX; ½Y; ZÞ gðY; ½X; ZÞþ gðZ; ½ X; YÞ; ð9:21Þ

where the ﬁrst line on the right hand side is understood according to (7.26).

A Riemannian connection is said to deﬁne a pseudo-Riemannian geometry on

a generalized Riemannian manifold with indeﬁnite metric tensor, it is said to

deﬁne a Riemannian geometry on a Riemannian manifold. A metric tensor

deﬁnes uniquely not only a (possibly indeﬁnite) metric on M, but also a (pseudo-)

Riemannian geometry. A tensor bundle or more generally any vector bundle

associated with LðMÞ with a Riemannian connection is called a Riemannian

vector bundle and g is called a Riemannian structure on it.

Put in local coordinates X ¼ o=ox

; Y ¼ o=ox

and Z ¼ o=ox

into (9.21). From

(3.19) it is immediately seen that the second line of (9.21) vanishes in this case.

Recall gðo=ox

; o=ox

Þ¼g

and, from (7.42), r

o=ox

ðo=ox

Þ¼

ðo=ox

Þ. One

readily obtains

308 9 Riemannian Geometry





¼ C

ikj

: ð9:22Þ

The last expression deﬁnes new Christoffel symbols for the Riemannian

connection. For historical reasons they are also called Christoffel symbols of the

ﬁrst kind while the general Christoffel symbols introduced in Sect. 7.9 are called

Christoffel symbols of the second kind. (Recall that Christoffel symbols are not

tensors.) For reference, some properties of Christoffel symbols valid for

Riemannian connections are given which are obvious from (9.22):





¼ C

; ð9:23Þ

ikj

¼ C

jki

;

¼ C

ijk

þ C

jik

: ð9:24Þ

Proof of the Fundamental Theorem Let a Riemannian connection be given and let

r be the corresponding covariant differential. Then, rg ¼ 0 and hT; X ^

Yi¼r

Y r

X ½X; Y¼0 (see (7.31)). From (7.27), 0 ¼ðrgÞðY; Z; XÞ¼

gðY; ZÞgðr

Y; ZÞgðY; r

ZÞ¼XgðY; ZÞgðr

Y; ZÞgðY; r

XÞg

ðY; ½X; ZÞ, since g is bilinear and gðY; ZÞ is a real function (see (7.26)). Likewise,

0 ¼r

gðX; ZÞgðr

X; ZÞgðX; r

ZÞ¼YgðX; ZÞgðr

Y; ZÞgðZ; ½Y; XÞ

gðX;r

ZÞ and 0 ¼r

gðX;YÞgðr

X;YÞgðX;r

YÞ¼ZgðX;YÞgðr

X;YÞ

gðX;r

ZÞgðX;½Z;YÞ. Adding the ﬁrst two relations and subtracting the last one

yields (9.21).

Conversely, assume that (9.21) is valid. Since g is non-degenerate, (9.21)

deﬁnes the action of r

on tangent vector ﬁelds Y uniquely, its action on functions

is understood according to (7.26) by deﬁnition. Hence, according to Sect. 7.7

the action of r

on tensor ﬁelds is uniquely deﬁned, if it is a derivative. Due to

multi-linearity of the right hand side of (9.21), (7.18) is readily fulﬁlled, and the

validity of (7.19) is easily checked. Hence, (9.21) deﬁnes a covariant derivative

with Christoffel symbols (9.23) for which the demanded transformation properties

(7.38) may straightforwardly calculated from the latter expression. This shows that

(9.21) deﬁnes uniquely a linear connection on M. With (9.24) and (7.43), rg ¼ 0

is straightforwardly calculated, and hence C is a metric connection. It is torsion

free which follows most easily from (7.47) and (9.23). h

The reader should perform the straightforward calculations of this proof as an

exercise.

Useful relations are obtained from





¼ g



9.3 Riemannian Connection 309

From g

¼ d

it follows that g

=ox

¼g

=ox

, and hence the ﬁrst term

of the displayed result is og

=ox

. Since ðg

Þ is the matrix inverse to ðg

Þ the

element g

is the minor of g

divided by det g,ordet gg

¼ o det g=og

and hence

o det g

¼ det gg

¼det gg

: ð9:25Þ

The last equality is just a special case of the preceding consideration. Now, one

has

¼



det g

o det g

¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj



: ð9:26Þ

On the other hand, rg ¼ 0 implies g

¼ 0, and with (7.44, 7.45) this means

og

=ox

¼ g

þ g

, which combines with (9.26)to

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj

: ð9:27Þ

The latter result yields for the divergence C

1;1

rX of a vector ﬁeld X

þ C

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj

oð

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

jdet gj

: ð9:28Þ

A similar expression is obtained for the divergence t

; j

of an alternating tensor t.

Let local coordinates be given in LðMÞ as before (7.23), and let it be provided

with a Riemannian connection with connection form x. Then, the ﬁrst structure

equation (7.23) for vanishing torsion reads (as previously, the index a of a

coordinate neighborhood is only occassionally used for the sake of clarity at local

forms and is dropped here)



^ x

¼ 0: ð9:29Þ

In view of (7.43), the condition for x to be a metric connection reads 0 ¼

rg ¼ðog

=ox

C

Þdx

dx

¼ðdg

g

Þdx

dx

where in the last equality the deﬁnition of the Christoffel symbols by the local

connection form given on p. 241 was used. Hence, this condition reads

 g

¼ 0; ð9:30Þ

where dg

is the ordinary differential of the component of the symmetric 2-form g.

These last two relations yield an equivalent formulation of the Fundamental

Theorem of Riemannian Geometry:

Given a generalized Riemannian manifold M with metric form g and given m

linearly independent 1-forms h

; i ¼ 1; ...; m on every coordinate neighborhood U

310 9 Riemannian Geometry

of M, there exists a unique set of m

1-forms x

solving the equations (9.29, 9.30)

and forming local connection forms of a Riemannian connection. The metric form

may then be expressed as g

 h

Use coordinate lines according to the integral curves of m standard horizontal

vector ﬁelds X

; hh

; X

i¼d

(p. 234). Then, g

¼ gðX

; X

Þ¼g

; X

ihh

; X

In the case of a Riemannian manifold (that is with a positive deﬁnite metric form)

one may choose linearly independent standard horizontal vector ﬁelds for which

¼ d

and hence ds

ðh

. Then, (9.30) yields a skew-symmetric local

connection form: x

þ x

¼ 0.

Since g

ðxÞ is a smooth function, the exterior derivative of its differential

vanishes: d ^ dg

¼ 0. The exterior derivative of (9.30) yields 0 ¼ g

þ g

^ x

þ g

^ x

þ g

^ x

þ g

. The second and the

fourth terms cancel, and, after some renaming of summation indices, ðx

^ x

Þg

þ g

ðx

^ x

þ dx

Þ¼0 results. According to (7.23), the parentheses of

the last relation contain the components of the curvature form X ¼ dx þ x ^ x,

so that this relation reads

þ g

¼ 0: ð9:31Þ

¼ X

of a Riemannian connection (with possibly indeﬁnite metric) is skew-

symmetric.

Expressing (7.30) in local coordinates results in (exercise, see also the text of

the next paragraph)

k\l

jkl

^ dx

; ð9:32Þ

where the curvature tensor R is given by (7.49). The curvature tensor ﬁeld of a

generalized Riemannian manifold,

ijkl

¼ g

jkl

; ð9:33Þ

has the following properties:

ijkl

¼R

jikl

¼R

ijlk

;

ijkl

þ R

iklj

þ R

iljk

¼ 0;

ijkl

¼ R

klij

ð9:34Þ

The ﬁrst line follows directly from (9.31, 9.32). To obtain the second line, ﬁrst

realize that (9.29) must hold locally for any linear independent system of 1-forms

. Putting locally h

¼ dx

yields 0 ¼

^ dx

. take the exterior derivative of

this relation and use it again to get 0 ¼

^ dx

ðX



^ x

Þ^

^ dx

. Insert (9.32) and obtain

j\k\l

jkl

^ dx

¼ 0 from

which the second line follows. Interchanging in the second line i with j, subtracting

9.3 Riemannian Connection 311

the result from the second line and observing the ﬁrst line yields 2R

ijkl

iklj

þ R

ljik

þ R

iljk

þ R

jkil

¼ 0. Interchanging here ij with kl, subtracting and

observing again the ﬁrst line of (9.34) yields the third one.

Finally, exterior differentiation of X ¼ dx þx ^ x yields the Bianchi identi-

ties in the form dX ¼ dx ^ x  x ^dx ¼ X ^ x  x ^X. This relation, which is

still a relation of forms on the frame bundle, in the notation of (7.30) paired with

tangent vectors X



on LðMÞ, may be pulled back from the canonical section to the

base manifold M as the same relation for the corresponding local forms on a

trivializing coordinate neighborhood in M, which are paired with tangent vectors

X on M as hX

; ðo=ox

Þ^ðo=ox

Þi ¼ R

jkl

ðo=ox

Þ or X

jkl

^ dx

(compare the text before (7.5) and before (7.14) as well as (7.46) and (9.32)).

Insert this expression together with x

from p. 241 into the relation dX

x

^ X

þ X

^ x

and obtain

jkl

^ dx

C

jkl

þ C

nkl



^ dx

Complete the left hand side to a covariant derivative according to (7.43) and get

jkl;m

^ dx

¼

jnl

þ C

jkn



^ dx

Now, this right hand side vanishes in the torsion free case, because according to

(7.47) in this case the Christoffel symbols are symmetric in the lower indices.

Hence, also the left hand side vanishes, which means that the corresponding

alternating combination of R

jkl;m

in the last three subscripts must vanish. This is the

sum with cyclic permutation minus the sum with anti-cyclic permutation of these

subscripts. In view of the alternating dependence of R on its last two subscripts the

six items can be combined into three. Thus,

ijkl;m

þ R

ijlm;k

þ R

ijmk;l

¼ 0 ð9:35Þ

expresses the Bianchi identities for a Riemannian connection.

9.4 Geodesic Normal Coordinates

In this section, the linear connection on M is specialized step by step.

First a general manifold M with a linear connection is considered, that is, the

frame bundle LðMÞ with a linear connection C. In a coordinate neighborhood

 M, the system of ordinary differential equations (7.51),

¼ 0; i ¼ 1; ...; m; ð9:36Þ

312 9 Riemannian Geometry

has the geodesics as solutions, which are the curves in M whose tangent vector is

transported parallel to itself along the curve. As is well known from standard

analysis, given initial conditions

ð0Þ¼u

;



¼ v

; ð9:37Þ

there are a neighborhood U  U

and two positive real numbers r; d, so that for all

ðu

Þ2U ¼ u

ðUÞ and ðv

Þ¼v 2 R

; jvj\r the system (7.51) has a unique

solution

¼ y

ðt; u

; v

Þ for jtj\d; ð9:38Þ

where y

depends smoothly on t; u

; v

. It may be thought of as a motion in time t

through M passing at t ¼ 0 through ðu

Þ with velocity vector ðv

Þ. Rescaling the

time by a factor c is equivalent to rescaling the velocity with 1=c:

ðct; u

; v

=cÞ¼y

ðt; u

; v

Þ or y

ðct; u

; v

Þ¼y

ðt; u

; cv

Þ: ð9:39Þ

In these relations, the left hand side is deﬁned where the right hand side was

deﬁned in (9.38). Take jcj\d and ﬁx ðu

Þ2U. Then, with the ﬁrst initial con-

dition (9.37), the relation

¼ y

ð1; u

; v

Þ with u

¼ y

ð0; u

; v

Þ¼y

ð1; u

; 0Þ¼u

ð9:40Þ

provides a mapping ðv

Þ7!ðx

Þ from a neighborhood of the origin of T

ðMÞR

onto a neighborhood of u 2 M with the origin of T

ðMÞ mapped to u. Indeed, now

with the second initial condition (9.37),

ð1; u

; v



v¼0

v ¼

ð1; u

; tv



t¼0

ðt; u

; v



t¼0

¼ v

and hence oy

ð1; u

; v

Þ=ov



v¼0

¼ d

so that the mapping is regular at v ¼ 0 and

provides a bijection of some neighborhood of v ¼ 0 onto a neighborhood of

u 2 M. This latter neighborhood is a coordinate neighborhood of M with coordi-

nates v

whose transition to the original coordinates x

is deﬁned by the ﬁrst

relation (9.40). The coordinates v

are called geodesic normal coordinates at u or

in short normal coordinates. Of course, they are determined up to the choice of a

base in T

ðMÞ, that is up to a non-degenerate linear transformation. (So far, M is a

linear connection space, a metric in M and angles in T

ðMÞ were not yet intro-

duced.) For every u 2 M and every ﬁxing of a base in T

ðMÞ there is a neigh-

borhood UðuÞ and a local coordinate system in UðuÞ of geodesic normal

coordinates at u; it is called a normal coordinate neighborhood of u.

Choose a coordinate neighborhood U

 M (with respect to coordinates x

), a

point u 2 U

and a base fX

ji ¼ 1; ...; mgin T

ðMÞ, and consider a tangent vector



. The points with geodesic normal coordinates ðv

¼ t



Þ form a

9.4 Geodesic Normal Coordinates 313

geodesic through the point u 2 M (at t ¼ 0) which has X

as its tangent vector at u.

If this tangent vector is parallel transported along the geodesic, it remains tangent to

it in all points of the geodesic (Fig. 9.1). From the above analysis it is clear that

given a point u

2 UðuÞ with normal coordinates v

(with respect to u), the tangent

vector X

of a geodesic from u to u

is uniquely deﬁned up to a scaling factor with an

inverse scaling of the curve parameter t of the geodesic. Such a rescaling does of

course not change the geodesic curve itself, it changes only its parametrization.

In a normal coordinate neighborhood UðuÞ of u 2 M, there is for every point

2 UðuÞ a uniquely deﬁned geodesic curve connecting u and u

For small enough q [ 0, there is a q-ball neighborhood

ðuÞ¼ x

i¼1

ðv

ðxÞÞ



()

; U

ðuÞ¼ x

i¼1

ðv

ðxÞÞ



()

ð9:41Þ

of u contained in the normal coordinate neighborhood UðuÞ (U

ðuÞ¼u

ðU

ðuÞÞ).

Moreover, if B

ð0Þ is the open ball of radius d centered at the origin of the

Euclidean space R

, then there is d [ 0 so that through every point u

2 U

ðuÞ and

for every v 2 B

ð0Þ there is a unique geodesic through u

with tangent vector v at u

and given by x

¼ y

ðt; u

; v

Þ; jtj\2. In summary, there is a mapping

w : U

ðuÞB

ð0Þ!U

ðuÞU

: ðu

; vÞ7!wðu

; vÞ¼ðu

; y

ð1; u

; v

ÞÞ:

ð9:42Þ

For every u 2 U

there are such positive numbers q and d (depending on u). Of

basic importance for the following is the statement visualized in Fig. 9.2:

For any point u in a linear connection manifold M there exists a neighborhood

W of u such that every point u

2 W has a normal coordinate neighborhood Uðu

that contains W.

Proof By the analysis after (9.39), the Jacobian of the mapping w is ½oðu

; y

Þ=

oðu

; v

Þ

ðu;0Þ

¼ 1. Hence, the differential w

ðu;0Þ



is a linear isomorphism, and from

the inverse function theorem (Sect. 3.5, case 3) it follows that there is a neighbor-

hood V of ðu; 0Þ2U

ðuÞB

ð0Þ and a positive number \d so that the restriction

Fig. 9.1 A geodesic through

the point u with the

corresponding geodesic

normal coordinates

314 9 Riemannian Geometry

:V ! U



ðuÞU



ðuÞ is a diffeomorphism. For every point u

2 U



ðuÞ take the

set of vectors B

¼fv 2 B



ð0Þjðu

; vÞ2Vg. Then, the set of points x 2 U

with

a-coordinates x

¼ y

ð1; u

; v

Þ; v 2 B

is contained in a normal coordinate

neighborhood Uðu

Þ of u

and is diffeomorphic to U



ðuÞ¼W: h

An immediate consequence is that every pair of points of W can be connected

by a geodesic.

Now, let the linear connection space W additionally be torsion-free. The normal

coordinates along a geodesic through a given point u may be chosen v

¼ t



hence d

=dt

¼ 0, with t ¼ 0 at u, and for instance unit vectors



v, and the system

(9.36)atu, expressed with the normal coordinates v

instead of the x

, reads

ð0Þ



¼ 0, where now the Christoffel symbols C

ðv

Þ refer to the normal

coordinates at u. Since in the torsion-free case these symbols are symmetric in the

lower indices in any coordinate system, it follows that C

ð0Þ¼0 for all

i; j; k ¼ 1; ...; m.

At any point u of a torsion-free linear connection manifold M, by choosing

geodesic normal coordinates at u the Christoffel symbols C

ð0Þ at u are made

vanish.

This is again an indication that the Christoffel symbols do not form a tensor, for

a tensor vanishing in some coordinate system would vanish in any coordinate

system.

Another remarkable property of torsion-free linear connection manifolds is that

the local connection form of a torsion-free linear connection manifold M is

uniquely deﬁned by the curvature tensor.

Proof Let u 2 M be any point and let UðuÞ be a normal coordinate neighborhood

of u. Fix a frame ðu; X

; ...; X

Þ at u and choose corresponding geodesic normal

coordinates in the following way:



;



¼ 1; is a tangent vector at u in

any direction given by the components



with respect to the ﬁxed frame; ðv

Þ¼

ðt



Þ are geodesic normal coordinates in UðuÞ on geodesics having the tangent

Fig. 9.2 See statement on

the previous page

9.4 Geodesic Normal Coordinates 315

vector



at u. Build a frame ﬁeld in UðuÞ by transporting the chosen frame at

u parallel (horizontally in LðMÞ) along geodesics. Tangent vectors to geodesics,

expressed in the polar coordinates



; t;

ðv

¼ 1 are o=ot and have constant

components



with respect to this frame ﬁeld along geodesics through u. Since the

frame ﬁeld is spanned by horizontal vectors, m linearly independent vector ﬁelds

with constant components



extend to standard horizontal vector ﬁelds Bð



Þ on

LðMÞ (compare (7.7)). Hence, according to (7.24), the canonical 1-form h and the

connection 1-form x are given as h



dt þ



; x



, where



and



depend

on the



and on t but do not contain dt (annihilate tangent vectors to geodesics



v ¼ const.). Moreover, since at t ¼ 0 a tangent vector in any direction is

proportional to dt (away from t ¼ 0 this holds only for vectors tangent to the

constructed geodesics through u),



t¼0

¼ 0 and



t¼0

¼ 0:

In view of the vanishing torsion, the structure equations (7.23) read

^ h

¼ 0; dx

^ x

¼ X

: ð9:43Þ

Since each component X

of the tensorial curvature form X is a horizontal 2-form,

it can be expressed as X

k\l



jkl

^ h

. In fact,



jkl

is the (bijective) u-trans-

formation of the curvature tensor R

jkl

¼ u



qrs

ðu

1

ðu

1

ðu

1

in the meaning of

(7.30) and (7.6); the u-transformation is not to be confused with the point u above.

The components of both tensors R and



R are functions of the



and of t. Comparing in

the equations (9.43) of forms only the terms proportional to dt yields







^ dt ¼ 0;







jkl



^ dt ¼ 0: ð9:44Þ

(Note that d



¼ dt ^ðo



=otÞþ¼ðo



=otÞ^dt þ:) Hence, the expres-

sions in parentheses must vanish. A further differentiation of the ﬁrst expression

with respect to t (t and



are independent) and then insertion of the second results in





jkl



jkl



¼ 0: ð9:45Þ

Given R

jkl

and hence



jkl

, this last equation is in fact an ordinary differential

equation for



, which with the initial conditions h

t¼0

¼ 0 and ðo



=otÞ

t¼0



t¼0

¼ d



has a unique solution



; tÞ¼



; tÞd



; tÞdt. (Recall that the components



are 1-forms, (9.45) is an equation of

forms and contains an equation for every



and every



Finally, take any point in UðuÞ with coordinates ðv

Þ¼ðt



Þ. For every ﬁxed

l ¼ 1; ...; m there is a (non-unique) regular linear transformation w with



. Inserting this into the ﬁrst equation (9.44) yields

316 9 Riemannian Geometry