Eschrig H. Topology and Geometry for Physics

Подождите немного. Документ загружается.

It was already seen in (8.84) that A is a local connection form of a Berry–Simon

connection and hence may be taken as a gauge potential. It is called a Mead–

Berry gauge potential. Correspondingly, the exterior covariant derivative

D ¼ 1r

þAðRÞð8:128Þ

is introduced which casts (8.125) into

D

=ð2lÞþEð RÞE1



ðRÞ¼0; ð8:129Þ

where W

ðRÞ means a column with elements W

ðRÞ, EðRÞ

¼ d

ðRÞ, matrix

multiplication is understood and (8.129) in principle contains an inﬁnite column of

equations, which in practical applications is cut off at a ﬁnite dimension for a small

number of lowest eigenvalues E

ðRÞ. The ordinary Born–Oppenheimer approxi-

mation means taking only the lowest E

ðRÞ and neglecting A.

Equations 8.127–8.129 form the basis of the gauge theory of molecular physics.

If the dimension of the problem (matrix dimension of A) is not cut of, the gauge

ﬁeld vanishes and the potential A can locally be gauged away. To see this, con-

sider the gauge ﬁeld F ¼ DA ¼ dAþA^Ain more detail:

iþ

ihW

^ dR

;

where R

and R

are local coordinates in a coordinate patch of the R-manifold M.

Because of orthonormality and completeness of the W, the second term in

parentheses is 

hðr

ÞjW

ihW

i ¼ hðr

Þjr

i. This may

be written as r

iþhW

i. Now, r

idR

¼ d

i¼0. Hence, F¼0 results. This does not mean that the full theory is

trivial. In fact, (8.127) is not deﬁned at points of term crossing, and hence all those

points have to be excluded from the manifold M, which thus becomes homotop-

ically highly non-trivial resulting in Aharonov–Bohm type situations. This is also

the main case of application of this gauge ﬁeld theory, if only a few lowest

electronic terms are retained. If there is a degeneracy of lowest electronic levels on

a whole manifold M, a case of non-Abelian gauge ﬁeld theory is realized. A wealth

of resulting phenomena is considered in [2].

References

1. Shapere, A., Wilczek, F. (eds.): Geometric Phases in Physics. World Scientiﬁc, Singapore

(1989)

2. Bohm, A., et al.: The Geometric Phase in Quantum Physics. Springer, Berlin (2003)

3. Schwarz, A.S.: Topology for Physicists. Springer-Verlag, Berlin (1994)

4. t’Hooft, G. (ed.): 50 Years of Yang-Mills Theory. World Scientiﬁc, Singapore (2005)

8.7 Gauge Field Theory of Molecular Physics 297

5. Milton, K.A.: Theoretical and experimental status of magnetic monopoles. Rep. Prog. Phys.

69, 1637–1711 (2006)

6. Nakahara, M.: Geometry, Topology and Physics. IOP Publishing, Bristol (1990)

7. Kato, T.: Perturbation Theory for Linear Operators. Springer, Berlin (1966)

8. Resta, R.: Macroscopic polarization in crystalline dielectrics: the geometric phase approach.

Rev. Mod. Phys. 66, 899–915 (1994)

9. Resta, R.: Manifestations of Berry’s phase in molecules and condensed matter. J. Phys.:

Condens. Matter 12, R107–R143 (2000)

10. Resta, R., Vanderbilt, D.: In: Ahn, C.H., Rabe, K.M., Triscone, J.M. (eds.). Physics of

Ferroelectrics: a Modern Perspective, pp. 31–68. Springer, Berlin (2007)

11. Eschrig, H.: The Fundamentals of Density Functional Theory. p. 226, Edition am

Gutenbergplatz, Leipzig, (2003)

12. Peierls, R.: Surprises in Theoretical Physics. Princeton University Press, Princeton (1979)

13. Qi, X.-L., Hughes, T.L., Zhang, S.-C.: Topological ﬁeld theory of timereversal invariant

insulators. Phys. Rev. B 78, 195424–1-43 (2008)

14. Ceresoli, D., Thonhauser, T., Vanderbilt, D., Resta, R.: Orbital magnetization in crystalline

solids: Multi-band insulators, Chern insulators, and metals. Phys. Rev. B74, 024408–1–13

(2006)

15. Hasan, M.Z., Kane, C.L.: Topological Insulators. Rev. Mod. Phys. 82, 3045–3067 (2010)

298 8 Parallelism, Holonomy, Homotopy and (Co)homology

Chapter 9

Riemannian Geometry

In Chaps. 3–8 the theory of manifolds was based on the smooth differentiable

structure, a complete atlas compatible with a pseudogroup S of class C

intro-

duced at the beginning of Sect. 3.1. With the only exception of Hodge’s star

operator introduced at the end of Sect. 5.1, a metric was not needed on general

manifolds and on bundles and was not introduced. Since the notion of manifold M

was restricted to local homeomorphy with R

in this text, by differentiation of real

functions along paths in M the tangent space was deﬁned in Sect. 3.3 as a local

linearization of M. On this basis, tensor bundles, the tensor calculus and the

exterior calculus as well as integration of exterior forms could be introduced and

the whole theory up to here could be built without a metric on M. Now, by deﬁning

a metric of a norm on the tangent spaces, due to the locally linear relation between

M and its tangent spaces, the manifold M itself is provided with a Riemannian

metric. A connection compatible with this metric makes M into a Riemannian

geometric space provided with a Riemannian geometry.

Despite the generality of Riemann’s concepts, in his time and afterwards, the

focus was on homogeneous manifolds having everywhere the same geometry, in

particular the same curvature. Only the work of Einstein and Hilbert on general

relativity provided an important application case of the concept of a general

Riemannian space. In recent time, with R. Hamilton’s concept of Ricci ﬂow which

is beyond the scope of this text, homogeneous manifolds came again into focus in

classiﬁcation problems of low-dimensional manifolds and in the theory of partial

differential equations. So far these developments culminated into Perelman’s proof

of Poincaré’s conjecture (see end of Sect. 2.5). Homogeneous manifolds are

considered in Sect. 9.2.

The Riemannian geometry proper, with the unique linear connection for which

everywhere rg ¼ 0, is treated in Sects. 9.3–9.5, while gravitation as the most

important application of this case is shortly considered in Sect. 9.6.

Apart from a few occasions, this short excursion through topology and

geometry was devoted to real manifolds. It is ﬁnally concluded with a brief

outlook on some complex generalizations.

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

DOI: 10.1007/978-3-642-14700-5_9, Ó Springer-Verlag Berlin Heidelberg 2011

299

9.1 Riemannian Metric

Let M be a manifold of dimension m, and let g be a symmetric tensor ﬁeld of type

ð0; 2Þ, in a coordinate neighborhood

g ¼ g

ðxÞdx

 dx

; g

¼ g

: ð9:1Þ

g is called a non-degenerate rank 2 tensor at point x 2 M, if the linear equation

system

ðxÞX

¼ 0; i ¼ 1; ...; m ð9:2Þ

has X

¼ 0 as its only solution, that is, det gðxÞ 6¼ 0. (For the sake of simplicity of

notation, both the tensor g and the matrix g ¼ðg

Þ in local coordinates are denoted

by the same letter.) g is called a positive deﬁnite rank 2 tensor at point x, if the

contraction

gðX; XÞ¼C

1;1

2;2

ðg  X  XÞ[0 for all X 6¼ 0; X 2 T

ðMÞ; ð9:3Þ

that is, the matrix ðg

ÞðxÞ is positive deﬁnite in the sense of linear algebra.

A generalized Riemannian manifold is a manifold M provided with an

everywhere on M non-degenerate symmetric tensor ﬁeld g of type ð0; 2Þ, its

metric tensor or fundamental tensor.Ifg is positive deﬁnite, then M is called a

Riemannian manifold.

The metric tensor of a Riemannian manifold M makes every tangent space

ðMÞ into a Euclidean space with inner product and norm

ðXjYÞ¼g

ðxÞX

; jXj¼ðXjXÞ

1=2

: ð9:4Þ

For the ﬁrst time in this text (besides mentioning it in passing in Sect. 5.1) this

deﬁnes an angle

\ðX; YÞ¼arccos

ðXjYÞ

jXjjYj



for j Xj 6¼ 0 6¼jYjð9:5Þ

between tangent vectors at the same point x.

Being a smooth tensor ﬁeld, (9.1) may be considered as a symmetric differ-

ential 2-form on M (to be distinguished from an exterior 2-form, which by

deﬁnition is alternating), called the metric form, in the following sense: Let

C : t

; t

½!M be a smooth path in M and let x ¼ CðtÞ for some t

\t\t

.Ina

coordinate neighborhood of x the path is given as x

¼ x

ðtÞ; i ¼ 1; ...; m. A tan-

gent vector to the path is X

¼ dx

=dt. By deﬁnition of a tensor ﬁeld (Sect. 4.1),

Besides an indeﬁnite metric there are many more generalizations of Riemannian manifold in

the mathematical literature; the case of an indeﬁnite metric is also called a pseudo-Riemannian

manifold. In this text generalized Riemannian manifold just comprises Riemannian and pseudo-

Riemannian manifold.

300 9 Riemannian Geometry

the expression (9.1) is independent of the choice of local coordinates. Hence, the

square of the norm of Xdt,

¼ g

; ð9:6Þ

is also independent of the choice of local coordinates, and the integral

sðCÞ¼

ﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃﬃ

dt ¼

ds ð9:7Þ

too has a real value independent of the used local coordinates. This integral is a

property of the path C alone and is called its arc length, ds ¼

ﬃﬃﬃﬃﬃﬃﬃ

is called the

element of the arc length.

Assigning in the case of a Riemannian manifold M this way an arc length to

every piecewise smooth curve is said to deﬁne a Riemannian metric on every

connected component of M. The corresponding distance function is

dðx; yÞ¼ inf

Cðx; yÞ

Cðx;yÞ

ds for g [ 0; ð9:8Þ

where Cðx; yÞ is any path from x to y in M, and the inﬁmum is taken over all paths.

As a distance function it must have the properties 1–3 given on p. 13. Let x be any

point of M and take any coordinate neighborhood U

of x. It is homeomorphic to a

neighborhood U

¼ u

ðU

ÞR

of x ¼ u

ðxÞ. As an open set of R

, for any

point y 6¼ x, U

contains a closed ball B

ðxÞ of some radius d [0 not containing

y ¼ u

ðyÞ. B

ðxÞ is compact, and hence the positive function of x equal to

min

fX;jXj¼1g

ðxÞX

of x takes on a positive minimum value there. Let this value

be e

. It is readily seen that dðx; yÞ[ de [ 0, no matter is y 2 U

or not: Cðx; yÞ\

is part of any path Cðx; yÞ and the above minimum value e yields an estimate of

the integral (9.8) over this part from below. Hence dðx; yÞ¼0, iff x ¼ y, and

property 1 from p. 13 is fulﬁlled. Property 2 is obvious, and property 3 simply

follows from concatenation of paths. Thus, (9.8) makes connected components of

M into metric spaces.

In the case of a generalized Riemannian manifold with g not being positive

deﬁnite, (9.6) is still called the element of the (sign carrying) arc length. Since in

every coordinate neighborhood the matrix g

is symmetric, it can be diagonalized

at a given point x 2 M, yielding ds

¼ g

ðdx

. Depending on the direction of dx

in M, ds

may be positive, zero or negative. Hence, in this case g does not deﬁne a

metric. Nevertheless, it is said to deﬁne an indeﬁnite Riemannian metric on M.

The expressions (9.4) are called an indeﬁnite inner product and an indeﬁnite

norm;(9.5) deﬁnes an angle for jXj 6¼ 0 6¼jYj, a non-zero vector X with jXj¼0 is

said to be isotropic. Although (9.8) does not make sense in this case, large parts of

the subsequent theory apply, and the indeﬁnite Minkowski metric gives it rele-

vance in physics.

9.1 Riemannian Metric 301

There exists a Riemannian metric on any m-dimensional smooth (paracompact)

manifold M.

Proof Recall that manifolds were supposed to be paracompact by deﬁnition.

Hence, there exists a locally ﬁnite coordinate covering fðU

; u

Þg of M and a

partition of unity fu

jsupp u

 U

; u

0g;

¼ 1 on M. Then, ds

i¼1

ðdx

deﬁnes a positive deﬁnite symmetric 2-form on U

aij

is the unit

matrix). Take a coordinate neighborhood U of x 2 M with x ¼ðx

Þ¼uðxÞ2

U  R

for which U is compact. It intersects with ﬁnitely many of the U

only.

Put

¼ g

; g

k¼1

The sum deﬁning g

is ﬁnite, and hence the deﬁnition is correct. Since the

Jacoby matrix ox

=ox

is regular and at least one of the u

ðxÞ, u

, say, is

positive, ds

ðxÞu

ðxÞ

ðdx

is positive deﬁnite and deﬁnes a Riemannian

metric on M: h

This statement means that the bundle of symmetric covariant tensors of rank 2

on every manifold M has a positive deﬁnite section. This is remarkable. Recall

from the end of Sect. 8.2, that the vector bundle over even a quite simple manifold,

while always having a section, may not have a non-zero section. In general, there

may also not exist an indeﬁnite Riemannian metric on M.

Let F : N ! M be an embedding of the manifold N into a Riemannian mani-

fold M with metric form g. Then,



gðX; YÞ¼gðF



X; F



YÞ for all X; Y 2 T

ðNÞð9:9Þ

deﬁnes a Riemannian metric on N. Such a statement does again not hold for an

indeﬁnite metric. (Why?)

The following considerations apply to generalized Riemannian manifolds with

both deﬁnite or indeﬁnite metric.

The transformation law of the metric form between overlapping coordinate

neighborhoods is

bij

akl

: ð9:10Þ

Denote the matrix inverse to g

by g

¼ d

: ð9:11Þ

Then, since the inverse of the Jacobi matrix ox

=ox

is ox

=ox

and since g

symmetric, from (9.10) it follows that

302 9 Riemannian Geometry

; ð9:12Þ

that is, (9.11) deﬁnes a symmetric contravariant rank 2 tensor with components g

(tensor of type ð2; 0Þ), uniquely attached with g.

Let x 2 M and X 2 T

ðMÞ. Then, x

¼ gðX;:Þ is a 1-form: hx

; Yi¼gðX; YÞ

is a real number for every Y 2 T

ðMÞ. Obviously, if X runs through the tangent

vector space T

ðMÞ, x

runs through the cotangent vector space T



ðMÞ. Since g is

non-degenerate, it provides a bijection between the tangent and cotangent vector

spaces at every point x 2 M, depending smoothly on x:IfX ¼ n

ðxÞðo=ox

Þ is a

tangent vector ﬁeld on U  M, then x

¼ðg

ðxÞn

ðxÞÞdx

is a cotangent vector

ﬁeld, since ðg

ðxÞn

ðxÞÞ depends smoothly on x and transforms like a cotangent

vector between overlapping coordinate neighborhoods, and X ¼ðg

ðo=ox

Þ¼ðg

Þðo=ox

Þ¼ðd

Þðo=ox

Þ¼n

ðo=ox

Þ:

The metric tensor g establishes an isomorphism between tangent and cotangent

spaces on M with its inverse mapping g

1

locally given by g

1

ðdx

; dx

Þ¼g

It likewise establishes an isomorphism between the spaces XðMÞ and D

ðMÞ of

tangent vector ﬁelds and cotangent vector ﬁelds (1-forms). Together with the

automorphism of the structure group which maps every group element to its

inverse, it also establishes an isomorphism between the tangent bundle TðMÞ and

the cotangent bundle T



ðMÞ.

The last of these statements is rather obvious. These isomorphisms extent by

linearity to isomorphisms between tensors, tensor ﬁelds and tensor bundles of types

ðr; sÞwith r þ s ﬁxed. In coordinate neighborhoods the corresponding mappings are

obtained by raising and lowering of tensor indices, for example, with some n,

...i

rþ1

...j

s1

¼ g

rþ1

...i

...j

n1

...j

s1

; t

...i

r1

...j

sþ1

¼ g

...i

n1

...i

r1

...j

: ð9:13Þ

Recall the convention (4.4) that in a tensor of type ðr; sÞ all contravariant indices

precede all covariant indices. If the tensor t is not symmetric, the order of rising or

lowering of indices must carefully be respected. Hence, in generalized Riemannian

manifolds insteadof types ðr; sÞof tensors only their rank r þs matters. Moreover, the

metric tensor g provides the following inner product in the tensor space of rank r þ s:

ðt juÞ¼t

...i

...j

g

...k

...l

: ð9:14Þ

In the case of an indeﬁnite Riemannian metric it is an indeﬁnite inner product.

9.2 Homogeneous Manifolds

Among the examples of principal ﬁber bundles at the end of Sect. 7.1, the notion of

homogeneous manifold was introduced as the quotient space G=H of a Lie group

G over its closed Lie subgroup H with the canonical projection p : G ! G=H.

9.1 Riemannian Metric 303

The homogeneous manifold forms the base space of the principal ﬁber bundle

ðG; G=H; p; HÞ.

A subgroup H of a Lie group G is not automatically a Lie subgroup. According

to the deﬁnition in Sect. 6.3, ðH; IdÞ must also be an embedded submanifold of G,

and that depends on the topology and differentiable structure introduced in H.

However,

if a manifold structure of the subgroup H of the Lie group G exists which makes

ðH; IdÞ into a (second countable) submanifold of G, it is unique, and H is a Lie

subgroup of G.

Proof Since H has a manifold structure, it has a tangent space T

ðHÞ at the

identity e 2 H  G. By left translations in G, pushed forward to tangent vectors, a

(dim H)-dimensional involutive distribution D on G is deﬁned (Sect. 3.6). Clearly,

at any h 2 H all tangent vectors of D

are in the tangent space T

ðHÞ. The con-

nected component H

is an integral manifold of D on G through e. Indeed: let

dim H ¼ k, and let for some h 2 H the tangent space T

ðHÞ be not contained in

. Then, there would be at least k þ1 curves through h in H, which are smooth in

G and have at least k þ 1 linearly independent tangent vectors in G. Left trans-

lating h back to e, the mapping ðx

; ...; x

kþ1

Þ7!ðF

ðx

Þ; ...; F

kþ1

ðx

kþ1

ÞÞ could be

completed by m k 1; m ¼ dim G, further linearly independent curves to a

diffeomorphism F of R

to some neighborhood U of e in G. F

1

 Id

would be a

smooth immersion of H \ U into R

containing some R

kþ1

. This is not possible:

H is second countable, and hence H \ U is an at most countable union of sets of

dimension k. Now, with the left translation l

; h 2 H in G, ðH; l

Þ is the uniquely

deﬁned (p. 78) smooth integral manifold of D through h and l

is a (smooth) left

translation of H. Moreover, ðh

; hÞ7!h

1

is smooth in G and hence smooth in

ðH; IdÞ. Since Id is injective, H is an embedded submanifold and a uniquely

deﬁned Lie subgroup of G: h

Hence, a subgroup of a Lie group can only in a unique way (with a uniquely

deﬁned total atlas) be a Lie subgroup under the identity mapping (inclusion

mapping). This answers uniqueness, it does not yet answer the question under

which conditions a submanifold structure exists making ðH; IdÞ into a Lie

subgroup of G.

From the above proof it can be seen that, if H is an embedded submanifold of G

in the relative topology from G, then it must be closed in G in this relative

topology. Conversely, let G be a second countable Lie group, and let H be a

subgroup of G closed in the relative topology from G. That implies that if U

is a

countable base of the topology of G, then H \ U

is a countable base of the relative

topology of H, and H is second countable. Let V be the closure of a coordinate

neighborhood U of e 2 G being homeomorphic to some closed ball of R

Then, H [ V is homeomorphic to a closed subset of this ball being complete in the

metric of R

and being a union of at most countably many closed subsets of H.

Hence, H [ V is a Baire space, and at least one of its subsets has an open interior

. Translating U

with all l

; h 2 H yields an embedding of H in G as a

304 9 Riemannian Geometry

submanifold with the relative topology. (A closed subset of R

in the relative

topology excludes the possibility of Example 5 on p. 76.)

ðH; IdÞ is a Lie subgroup of the (second countable) Lie group G, iff it is a

subgroup closed in the relative topology from G.

Hence, if G is a (second countable) Lie group and H is a closed subgroup of G

(in the relative topology), then ðG; G=H; p; HÞ with the uniquely deﬁned manifold

structure of H as above is a principal ﬁber bundle, and G=H is a homogeneous

manifold.

Let G be a Lie group acting smoothly from the left on a manifold M by

R : G M ! M. Then, for a ﬁxed g 2 G, Rðg; Þ ¼ R

: M ! M is obviously a

diffeomorphism of M into itself. Pick m 2 M, then G

¼fg 2 G jR

ðmÞ¼mg is a

closed (as the preimage of the closed set fmg) subgroup of G, the isotropy group

at m. (It may be trivial: G

¼feg.) G

acts also from the left on M by R

¼ Rj

and leaves m on place as a ﬁxed point. By linearization, this action can be pushed

forward to a mapping R

m

: G

! AutðT

ðMÞÞ of the isotropy group into the

automorphism group of the tangent space at m, yielding a Lie group representation

of G

in the vector space T

ðMÞ. (Exercise: Show that R

m

is a smooth

homomorphism of Lie groups.) The image of the homomorphism R

m

is called the

linear isotropy group at m.

Consider as an example the Lie group OðnÞ of real orthogonal ðn  nÞ-matrices

acting from the left on the unit sphere S

n1

of R

. The elements of OðnÞ of the form

R ¼



; ð9:15Þ

where R

is an element of Oðn 1Þ and the zeros denote zero column and row, are

precisely the elements of the isotropy group at the south pole s ¼ð0; ...; 0; 1Þ of

n1

, and Oðn  1Þ is a closed subgroup of OðnÞ (exercise). Since T

ðS

n1

Þ

n1

, the linear isotropy group at s is again Oðn 1Þ in this case.

Now, let R be a transitive action of the Lie group G on M (p. 206), and let again

m 2 M. Then, the mapping

R : G=G

! M : gG

RðgG

Þ¼R

ðmÞð9:16Þ

is correctly deﬁned, since for every g

2 G

one has R

ðmÞ¼R

ðR

ðmÞÞ ¼

ðmÞ:

R is onto and one–one: it is surjective since G acts transitively, and it is

easily shown that

Rðg

Þ¼

Rðg

Þ implies g

1

2 G

and hence

R is

injective. It can even be shown [1] that it is a diffeomorphism and hence M and

G=G

are equivalent homogeneous manifolds.

Returning to the above example G ¼ OðnÞ; M ¼ S

n1

; G

¼ Oðn  1Þ, the

quotient space OðnÞ=Oðn  1Þ consists of the cosets ROðn  1Þ; R 2 OðnÞ of

the subgroup Oðn 1Þ in OðnÞ. There is the diffeomorphism OðnÞ=

Oðn  1Þ!S

n1

: ROðn  1Þ7!Rs; R 2 OðnÞ, and OðnÞ=Oðn 1Þ and S

n1

are

equivalent homogeneous manifolds. As was shown on p. 194, OðnÞ consists of two

connected components for det R ¼1, and OðnÞ

¼ SOð nÞ. Therefore one has also

9.2 Homogeneous Manifolds 305

an equivalence of the former two homogeneous manifolds with SOðnÞ=SOðn 1Þ,

which was already considered in Sect. 2.6. Consider SOð3Þ, the group of all

rotations of R

. If one ﬁxes the south pole of the unit sphere S

in R

, then there

remain the rotations with the rotation axis through the south pole ﬁxed, which form

the group SOð2Þ of rotations of the equator of the sphere S

the elements of which

are just given by the angle of rotation. Any coset RSOð2Þ; R 2 SOð3Þ consists of a

rotation with the south pole ﬁxed and a subsequent free rotation, which possibly

moves the south pole to any point of the sphere S

. This can also be achieved by ﬁrst

rotating the south pole to the new position and then making a rotation with that

position ﬁxed. (R and the element of SOð2Þ related to the new axis are of course

different in this case, since SOð3Þ is non-Abelian.) Hence, all cosets of

SOð3Þ=SOð2Þ are in one–one and onto correspondence with all points of the sphere

. All those points can be obtained by an SOð3Þ-rotation Rs of the south pole s.

Let G be a (second countable) Lie group and let H be its invariant closed

subgroup. Then the homogeneous manifold G=H with its quotient group structure

is a Lie group.

Since G= H is a manifold, it is only to check that the group operations are

smooth. This is straightforward by use of local coordinates.

Now, let G be a (second countable) Lie group and let H be a compact subgroup

(in the relative topology). Then, H is a Lie subgroup and G=H is a homogeneous

manifold of cosets x ¼ g

H; g

2 G (g

deﬁnes uniquely a coset x, but not vice

versa) on which G acts transitively from the left by G G=H ! G=

H : ðg; g

HÞ7!gg

H. Pick x 2 G=H. The isotropy group at x is G

¼fg 2 G j

gðg

HÞ¼g

Hg which implies g

1

H ¼ H and, since cosets are disjoint,

1

2 H. It is not difﬁcult to see (exercise) that together with H also G

and the

linear isotropy group R

x

of transformations of the tangent space T

ðG=HÞ are

compact. In a compact group, a ﬁnite invariant measure (Haar’s measure) can be

introduced. Take any positive scalar product in the vector space T

ðG=HÞ and

average it over the invariant measure with respect to the transformations of R

x

The result is an invariant scalar product g

ðX; YÞ¼g

ðxÞX

¼ g

ðxÞðg



XÞ

ðg



YÞ

¼ g

ðg



X; g



YÞ for all X; Y 2 T

ðG=HÞ and all g



2 R

x

. Since G acts

transitively on G=H, for every x

2 G=H there is g

2 G so that x

¼ g

x. Then,

ðX; YÞ¼g

ðg

1

0

X; g

1

0

YÞð9:17Þ

is easily shown to be independent of the special choice of g

. It deﬁnes an

invariant metric on the homogeneous manifold G=H and makes this manifold

into a homogeneous Riemannian manifold.

In the above example this is just the ordinary metric on the sphere S

n1

which in

orthogonal local coordinates is given by the unit matrix g.

Let again G be a Lie group and consider the product manifold G G (not the

direct product of groups) with the group multiplication ðg

; g

Þðg

; g

Þ¼

ðg

; g

Þ. It is easily seen that this makes G G into another Lie group.

Consider its action on G from the left as ðG  GÞG ! G : ððg

; g

Þ; gÞ7!

306 9 Riemannian Geometry