Назад
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
R
þAðRÞð8:128Þ
is introduced which casts (8.125) into
D
2
=ð2lÞþEð RÞE1

W
E
ðRÞ¼0; ð8:129Þ
where W
E
ðRÞ means a column with elements W
a
E
ðRÞ, EðRÞ
a
b
¼ d
a
b
E
a
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
a
ðRÞ. The ordinary Born–Oppenheimer approxi-
mation means taking only the lowest E
a
ðRÞ and neglecting A.
Equations 8.1278.129 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:
F
a
b
¼
X
ij
r
R
i
hW
a
jr
R
j
jW
b
X
c
hW
a
jr
R
i
jW
c
ihW
c
jr
R
j
jW
b
i
!
dR
i
^ dR
j
;
where R
i
and R
j
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
P
c
hðr
R
i
W
a
ÞjW
c
ihW
c
jr
R
j
jW
b
i ¼ hðr
R
i
W
a
Þjr
R
j
jW
b
i. This may
be written as r
R
i
hW
a
jr
R
j
jW
b
iþhW
a
jr
R
i
r
R
j
jW
b
i. Now, r
R
i
r
R
j
jW
b
idR
i
^
dR
j
¼ d
2
jW
b
0. Hence, 0 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. 38 the theory of manifolds was based on the smooth differentiable
structure, a complete atlas compatible with a pseudogroup S of class C
1
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
m
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. 9.39.5, 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
ij
ðxÞdx
i
dx
j
; g
ij
¼ g
ji
: ð9:1Þ
g is called a non-degenerate rank 2 tensor at point x 2 M, if the linear equation
system
g
ij
ðxÞX
j
¼ 0; i ¼ 1; ...; m ð9:2Þ
has X
j
¼ 0 as its only solution, that is, det gðxÞ 0. (For the sake of simplicity of
notation, both the tensor g and the matrix g ¼ðg
ij
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
C
2;2
ðg X XÞ[0 for all X 0; X 2 T
x
ðMÞ; ð9:3Þ
that is, the matrix ðg
ij
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.
1
The metric tensor of a Riemannian manifold M makes every tangent space
T
x
ðMÞ into a Euclidean space with inner product and norm
ðXjYÞ¼g
ij
ðxÞX
i
Y
j
; 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 0 6¼jY9:5Þ
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
0
; t
1
½!M be a smooth path in M and let x ¼ CðtÞ for some t
0
\t\t
1
.Ina
coordinate neighborhood of x the path is given as x
i
¼ x
i
ðtÞ; i ¼ 1; ...; m. A tan-
gent vector to the path is X
i
¼ dx
i
=dt. By definition of a tensor field (Sect. 4.1),
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,
ds
2
¼ g
ij
dx
i
dt
dx
j
dt
dt
2
¼ g
ij
dx
i
dx
j
; ð9:6Þ
is also independent of the choice of local coordinates, and the integral
sðCÞ¼
Z
t
1
t
0
ffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffiffi
g
ij
dx
i
dt
dx
j
dt
r
dt ¼
Z
C
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
ffiffiffiffiffiffi
ds
2
p
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
Z
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
a
of x. It is homeomorphic to a
neighborhood U
a
¼ u
a
ðU
a
ÞR
m
of x ¼ u
a
ðxÞ. As an open set of R
m
, for any
point y x, U
a
contains a closed ball B
d
ðxÞ of some radius d [0 not containing
y ¼ u
a
ðyÞ. B
d
ðxÞ is compact, and hence the positive function of x equal to
min
fX;jX1g
g
ij
ðxÞX
i
X
j
of x takes on a positive minimum value there. Let this value
be e
2
. It is readily seen that dðx; yÞ[ de [ 0, no matter is y 2 U
a
or not: Cðx; yÞ\
U
a
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
ij
is symmetric, it can be diagonalized
at a given point x 2 M, yielding ds
2
¼ g
ii
ðdx
i
Þ
2
. Depending on the direction of dx
i
in M, ds
2
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 0 6¼jYj, a non-zero vector X with jX0 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 U
a
; u
a
Þg of M and a
partition of unity fu
a
jsupp u
a
U
a
; u
a
0g;
P
a
u
a
¼ 1 on M. Then, ds
2
a
¼
P
m
i¼1
ðdx
i
a
Þ
2
defines a positive definite symmetric 2-form on U
a
(g
aij
is the unit
matrix). Take a coordinate neighborhood U of x 2 M with x ¼ðx
i
Þ¼uðxÞ2
U R
m
for which U is compact. It intersects with finitely many of the U
a
only.
Put
ds
2
¼ g
ij
dx
i
dx
j
; g
ij
¼
X
a
X
m
k¼1
u
a
ox
k
a
ox
i
ox
k
a
ox
j
:
The sum defining g
ij
is finite, and hence the definition is correct. Since the
Jacoby matrix ox
k
a
=ox
i
is regular and at least one of the u
a
ðxÞ, u
b
, say, is
positive, ds
2
ðxÞu
b
ðxÞ
P
i
ðdx
i
b
Þ
2
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
x
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
g
bij
¼
ox
k
a
ox
i
b
g
akl
ox
l
a
ox
j
b
: ð9:10Þ
Denote the matrix inverse to g
ij
by g
ij
:
X
k
g
ik
g
kj
¼
X
k
g
jk
g
ki
¼ d
i
j
: ð9:11Þ
Then, since the inverse of the Jacobi matrix ox
k
a
=ox
j
b
is ox
k
b
=ox
j
a
and since g
kl
is
symmetric, from (9.10) it follows that
302 9 Riemannian Geometry
g
ij
b
¼
ox
i
b
ox
k
a
g
kl
a
ox
j
b
ox
l
a
; ð9:12Þ
that is, (9.11) defines a symmetric contravariant rank 2 tensor with components g
ij
(tensor of type ð2; 0Þ), uniquely attached with g.
Let x 2 M and X 2 T
x
ðMÞ. Then, x
X
¼ gðX;:Þ is a 1-form: hx
X
; YgðX; YÞ
is a real number for every Y 2 T
x
ðMÞ. Obviously, if X runs through the tangent
vector space T
x
ðMÞ, x
X
runs through the cotangent vector space T
x
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
i
ðxÞðo=ox
i
Þ is a
tangent vector field on U M, then x
X
¼ðg
ij
ðxÞn
i
ðxÞÞdx
j
is a cotangent vector
field, since ðg
ij
ðxÞn
i
ðxÞÞ depends smoothly on x and transforms like a cotangent
vector between overlapping coordinate neighborhoods, and X ¼ðg
ik
x
Xk
Þ
ðo=ox
i
Þ¼ðg
ik
g
kl
n
l
Þðo=ox
i
Þ¼ðd
i
l
n
l
Þðo=ox
i
Þ¼n
i
ðo=ox
i
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
i
; dx
j
Þ¼g
ij
.
It likewise establishes an isomorphism between the spaces MÞ and D
1
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,
t
i
1
...i
rþ1
j
1
...j
s1
¼ g
i
rþ1
k
t
i
1
...i
r
j
1
...j
n1
kj
n
...j
s1
; t
i
1
...i
r1
j
1
...j
sþ1
¼ g
j
1
k
t
i
1
...i
n1
ki
n
...i
r1
j
2
...j
s
: 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
1
...i
r
j
1
...j
s
g
j
1
l
1
g
j
s
l
s
g
i
1
k
1
g
i
r
k
r
u
k
1
...k
r
l
1
...l
s
: 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
e
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
h
are in the tangent space T
h
ðHÞ. The con-
nected component H
e
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
h
ðHÞ be not contained in
D
h
. 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
1
; ...; x
kþ1
Þ7F
1
ðx
1
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
m
to some neighborhood U of e in G. F
1
Id
H
would be a
smooth immersion of H \ U into R
m
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
; h 2 H in G, ðH; l
h
Þ is the uniquely
defined (p. 78) smooth integral manifold of D through h and l
h
is a (smooth) left
translation of H. Moreover, ðh
0
; hÞ7!h
0
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
n
is a
countable base of the topology of G, then H \ U
n
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
m
.
Then, H [ V is homeomorphic to a closed subset of this ball being complete in the
metric of R
m
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
U
H
. Translating U
H
with all l
h
; 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
2
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
g
: M ! M is obviously a
diffeomorphism of M into itself. Pick m 2 M, then G
m
¼fg 2 G jR
g
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
m
¼feg.) G
m
acts also from the left on M by R
m
¼ Rj
G
m
and leaves m on place as a fixed point. By linearization, this action can be pushed
forward to a mapping R
m
: G
m
! AutðT
m
ðMÞÞ of the isotropy group into the
automorphism group of the tangent space at m, yielding a Lie group representation
of G
m
in the vector space T
m
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
n1
of R
n
. The elements of OðnÞ of the form
R ¼
R
0
0
01

; ð9:15Þ
where R
0
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
S
n1
, and Oðn 1Þ is a closed subgroup of OðnÞ (exercise). Since T
s
ðS
n1
Þ
R
n1
, 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
! M : gG
m
7!
~
RðgG
m
Þ¼R
g
ðmÞð9:16Þ
is correctly defined, since for every g
0
2 G
m
one has R
gg
0
ðmÞ¼R
g
ðR
g
0
ðmÞÞ ¼
R
g
ðmÞ:
~
R is onto and one–one: it is surjective since G acts transitively, and it is
easily shown that
~
Rðg
1
G
m
Þ¼
~
Rðg
2
G
m
Þ implies g
1
2
g
1
2 G
m
and hence
~
R is
injective. It can even be shown [1] that it is a diffeomorphism and hence M and
G=G
m
are equivalent homogeneous manifolds.
Returning to the above example G ¼ OðnÞ; M ¼ S
n1
; G
s
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
n1
: ROðn 1Þ7!Rs; R 2 OðnÞ, and OðnÞ=Oðn 1Þ and S
n1
are
equivalent homogeneous manifolds. As was shown on p. 194, OðnÞ consists of two
connected components for det R ¼1, and OðnÞ
e
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
3
. If one fixes the south pole of the unit sphere S
2
in R
3
, 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
2
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
2
. 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
S
2
. 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
x
H; g
x
2 G (g
x
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
x
HÞ7!gg
x
H. Pick x 2 G=H. The isotropy group at x is G
x
¼fg 2 G j
gðg
x
HÞ¼g
x
Hg which implies g
1
x
gg
x
H ¼ H and, since cosets are disjoint,
g
1
x
gg
x
2 H. It is not difficult to see (exercise) that together with H also G
x
and the
linear isotropy group R
x
of transformations of the tangent space T
x
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
x
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
x
ðX; YÞ¼g
ij
ðxÞX
i
Y
j
¼ g
ij
ðxÞðg
XÞ
i
ðg
YÞ
j
¼ g
x
ðg
X; g
YÞ for all X; Y 2 T
x
ðG=HÞ and all g
2 R
x
. Since G acts
transitively on G=H, for every x
0
2 G=H there is g
xx
0
2 G so that x
0
¼ g
xx
0
x. Then,
g
x
0
ðX; YÞ¼g
x
ðg
1
xx
0
X; g
1
xx
0
YÞð9:17Þ
is easily shown to be independent of the special choice of g
xx
0
. 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
n1
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
1
; g
2
Þðg
0
1
; g
0
2
Þ¼
ðg
1
g
0
1
; g
2
g
0
2
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
1
; g
2
Þ; gÞ7!
306 9 Riemannian Geometry