Basic Material 407
If
λ
= 0andg
λ
= {0 },then
λ
is called a restricted root and g
λ
a restricted root space
of g with respect to a. We denote by Ψ the set of all restricted r oots of g with respect
to a.Therestricted root space decomposition of g is the direct sum decomposition
g = g
0
⊕
λ
∈Ψ
g
λ
.
We always have
[g
λ
,g
μ
] ⊂ g
λ
+
μ
and
θ
(g
λ
)=g
−
λ
for all
λ
,
μ
∈Ψ. Moreover,
g
0
= k
0
⊕a,
where k
0
is the centralizer of a in k. We now choose a notion of positivity for a
∗
,
which leads to a subset Ψ
+
of positive restricted roots. Then
n =
λ
∈Ψ
+
g
λ
is a nilpotent subalgebra of g. Any two such nilpotent subalgebras are conjugate via
Ad(k) for some k in the normalizer of a in K. The vector space direct sum
g = k ⊕a ⊕n
is called a n Iwasawa decomposition of g. The vector space s = a ⊕n is, in fact, a
solvable subalgebra of g with [s,s]=n.LetA, N be the connected Lie subgroups of
G with Lie alg ebras a,n, respectively. Then A and N are simply connected and the
map
K ×A ×N →G , (k,a,n) → kan
is a diffeomorphism onto G, a so-called Iwasawa decomposition of G.
If t is a maximal abelian subalgebra of k
0
,thenh = t⊕a is a Cartan subalgebra of
g,thatis,h(C) is a Cartan subalgebra of g(C). Consider the root space decomposition
g(C)=h(C)⊕
α
∈Δ
(g(C))
α
of g(C) with respect to h(C).Thenwehave
g
λ
= g ∩
α
∈Δ,
α
|a=
λ
(g(C))
α
for all
λ
∈Ψ and
(k
0
)(C)=t(C) ⊕
α
∈Δ,
α
|a=0
(g(C))
α
.
In particular, all roots are real on it ⊕a. Of particular interest are those real forms of
g(C) for which a is a Cartan subalgebra of g. In this case g is called a split real form
408 Submanifolds and Holonomy
of g(C). Note that g is a split real form if and only if k
0
, the centralizer of a in k,is
trivial. The split real f orms of the complex simple Lie algebras are, for the classical
complex Lie algebras,
sl
r+1
(R) ⊂ sl
r+1
(C) , so
r,r+1
⊂ so
2r+1
(C) , sp
r
(R) ⊂ sp
r
(C) , so
r,r
⊂ so
2r
(C) ,
and, for the exceptional complex Lie algebras,
e
6
6
⊂ e
6
(C) , e
7
7
⊂ e
7
(C) , e
8
8
⊂ e
8
(C) , f
4
4
⊂ f
4
(C) , g
2
2
⊂ g
2
(C) .
A.3 Homogeneous spaces
A homogeneous space is a manifold with a transitive group of transformations.
Homogeneous spaces provide excellent examples for studying the interplay between
analysis, geometry, algebra and topology. A modern introduction to homogeneous
spaces can be found in Kawakubo [167]. Further results on Lie transformation groups
can be found in [270].
The quotient space G/K
Let G be a Lie group and K be a closed subgroup of G.ByG/K we denote the
set of left cosets of K in G,
G/K = {gK : g ∈ G },
and by
π
the canonical projection
π
: G → G/K , g → gK.
We equip G/K with the quotient topology relative to
π
.Then
π
is a continuous
map and, since K is closed in G, the coset space G/K is a Hausdorff space. There is
exactly one smooth manifold structure on G/K (which is even real analytic) so that
π
becomes a smooth map and local smooth sections of G/K in G exist. If K is a normal
subgroup of G,thenG/K becomes a Lie group with respect to the multiplication
g
1
K ·g
2
K =(g
1
g
2
)K.
If K is a closed subgroup of a Lie group G,then
G ×G/K →G/K , (g
1
,g
2
K) → (g
1
g
2
)K
is a transitive smooth action of G on G /K. In fact, the smooth structure on G/K can
be characterized by the property that this action is smooth. Conversely, suppose we
have a transitive smooth action
G ×M → M , (g, p) → gp
Basic Material 409
of a Lie group G on a smooth manifold M.Letp be a point in M and
G
p
= {g ∈ G : gp = p}
the isotropy group of G at p.Ifq is another point in M and g ∈ G with gp = q,then
G
q
= gG
p
g
−1
. Thus the isotropy groups of G are all conjugate to each other. The
isotropy group G
p
is obviously closed in G. Thus we can equip G/G
p
with a smooth
manifold structure as described above. With respect to this structure, the map
G/G
p
→ M , gG
p
→ gp
is a smooth diffeomorphism. In this way we will always identify the smooth manifold
M with the coset space G/K. Moreover,
π
: G →G/K is a principal fiber bundle with
fiber and structure group K,whereK acts on G by multiplication from the right.
Homogeneous spaces
If M is a smooth manifold and G is a Lie group acting transitively on M,we
say that M is a homogeneous space, or, more precisely, a homogeneous G-space.
If M is a connected homogeneous G-space, then the identity component G
o
of G
also acts transitively on M. This allows us to reduce many problems on connected
homogeneous spaces to connected Lie groups and thereby to Lie algebras. Another
important fact, proved by Montgomery, is that, if M = G/K is a compact homoge-
neous G-space with G and K connected, then there exists a compact subgroup of G
acting transitively on M. This makes it possible to use the many useful features of
compact Lie groups for studying compact homogeneous spaces.
Effective a ctions
Let M be a homogeneous G-space and
φ
: G → Diff(M) be the homomorphism
from G into the diffeomorphism group of M assigning to each g ∈ G the diffeomor-
phism
ϕ
g
: M → M , p → gp.
The action of G on M is said to be effective if ker(
φ
)={e},wheree denotes the
identity in G. In other words, an action is effective if just the identity of G actsasthe
identity transformation on M. Writing M = G /K, we can characterize ker(
φ
) as the
largest normal subgroup of G that is contained in K. Thus, G/ker(
φ
) is a Lie group
with an effective transitive action on M.
Reductive decompositions
Let M = G/K be a homogeneous G-space. We denote by e the identity of G and
put o = eK ∈ M.Letg and k be the Lie algebras of G and K, respectively. As usual,
we identify the tangent space of a Lie group at the identity with the corresponding
Lie algebra. We choose any linear subspace m of g complementary to k,sothat
g = k ⊕m. Then the differential d
e
π
at e of the projection
π
: G → G/K gives rise to
an isomorphism
d
e
π
|
m
: m → T
o
M.
410 Submanifolds and Holonomy
One of the basic tools in studying homogeneous spaces is to use this isomorphism
to identify tangent vectors of M at o with elem ents in the Lie algebra g.However,
there are many choices of complementary subspaces m, and some turn out to be more
useful than others. We will describe this now.
Let Ad : G →GL(g) be the adjoint representation of G. The subspace m is said to
be Ad(K)-invariant if Ad(k)m ⊂ m for all k ∈ K.Ifm is Ad(K)-invariant and k ∈ K,
the differential d
o
ϕ
k
at o of the diffeomorphism
ϕ
k
: M → M , p →kp has the simple
expression
d
o
ϕ
k
= Ad(k)|
m
.
For this reason one is interested in finding Ad(K)-invariant linear subspaces m of
g. Unfortunately, not every homogeneous space admits such subspaces. A homoge-
neous space G/K is called reductive if there is an Ad(K)-invariant linear subspace m
of g so that g = k ⊕m. In this situation, g = k ⊕m is called a reductive decompositio n
of g.
Isotropy representations and invariant metrics
The homomorphism
χ
: K → GL(T
o
M) , k → d
o
ϕ
k
is called the isotropy representation of the homogeneous space G/K and the image
χ
(K) ⊂GL(T
o
M) is called the linear isotropy group of G/K.IfG/K is reductive and
g = k⊕m is a reductive decomposition, the isotropy representation of G/K coincides
with the adjoint representation
Ad|
K
: K → GL(m)
(via the identification m = T
o
M).
The linear isotropy group contains the information for deciding whether or not a
homogeneous space G/K can be equipped with a G-invariant Riemannian structure.
A G-invariant Riemannian metric ·, · on M = G/K is a Riemannian metric so that
ϕ
g
is an isometry of M for each g ∈ G,thatis,ifG acts on M by isometries. A ho-
mogeneous space M = G/K can be equipped with a G-invariant Riemannian metric
if and only if the linear isotropy group
χ
(K) is a relatively compact subset of the
topological space L(T
o
M,T
o
M) of all linear maps T
o
M → T
o
M. It follows that every
homogeneous space G/K with K compact admits a G-invariant Riemannian metric.
Every Riemannian homogeneous space is reductive. If G/K is re ductive and
g = k⊕m is a reductive decomposition, then there is a one-to-one correspondence be-
tween the G-invariant Riemannian metrics o n G/K and the positive definite Ad(K)-
invariant symmetric bilinear forms on m. Any such bilinear form defines a Riemann-
ian metric on M by requiring that each
ϕ
g
is an isometry. The Ad (K)-invariance of
the bilinear form ensures that the inner product o n each tangent space is well de-
fined. In particular, if K = {e},thatis,ifM = G is a Lie group, then the G-invariant
Riemannian metrics on M are exactly the left-invariant Riemannian metrics on G.
Basic Material 411
We finally remark that, as a consequence of Schur’s Lemma, a G-invariant Riemann-
ian metric on a homogeneous space G/K is unique up to homothety if the isotropy
representation is irreducible.
Naturally reductive Riemannian homogeneous spaces
A homogeneous Riemannian manifold M is said to be a naturally reductive Rie-
mannian homogeneous space if there exists a connected Lie subgroup G of the isom-
etry group I(M) of M acting transitively and effectively o n M and a reductive de-
composition g = k ⊕m of the Lie algebra g of G,wherek is the Lie algebra of the
isotropy group K of G at some point o ∈ M, such that
[X, Z]
m
,Y + Z,[X,Y ]
m
= 0
for all X ,Y,Z ∈ m,where·,· denotes the inner product on m that is induced b y the
Riemannian metric on M and [·,·]
m
denotes the canonical projection onto m with re-
spect to the decomposition g = k ⊕m . Any such decomposition is called a naturally
reductive decomposition of g. The above algebraic condition is equivalent to the ge-
ometric property that every geodesic
γ
: R → M in M with
γ
(0)=o can be realized
as
γ
(t)=Exp(tX)(o) for some X ∈ m.
A.4 Riemannian symmetric spaces and flag manifolds
Riemannian symmetric spaces form an important subclass of the homogeneous
spaces and were studied and classified by
´
Elie Cartan. The fundamental books on this
topic are Helgason [151] and Loos [201]. Another nice introduction can be found in
[158]. Flag manifolds are homogeneous spaces that are closely related to symmetric
spaces.
(Locally) symmetric spaces
Let M be a Riemannian manifold, p ∈ M,andr ∈R
+
be sufficiently small so that
normal coordinates are defined on the open ball B
r
(p) consisting of all points in M
with distance less than r to p. Denote by exp
p
: T
p
M → M the exponential map of M
at p.Themap
s
p
: B
r
(p) → B
r
(p) , exp
p
(v) → exp
p
(−v)
reflects in p the geodesics of M through p and is called a local geodesic symmetry at
p. A connected Riemannian manifold is called a Riemannian locally symmetric space
if at each point p in M there exists an open ball B
r
(p) such that the corresponding
local geodesic symmetry s
p
is an isometry. A connected Riemannian manifold is
called a Riemannian symmetric space if at each point p ∈ M such a local geodesic
symmetry extends to a global isometry s
p
: M → M. This is equivalent to saying that
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