256 Submanifolds and Holonomy
and let F
p
be the horosphere containing p. Denote by
˜
G the identity component of
the subgroup of G leaving F
p
invariant. Then
˜
G is transitive on F
p
.LetH
n
= G/H
be naturally reductive with respect to th e decomposition g = h ⊕m, m T
p
M,and
identify m
0
T
p
F
p
,wherem
0
⊂m. Then, if v ∈m
0
,Exp(tX)p is a geodesic tangent
to F
p
. Since the foliation F is invariant under G,wegetExp(tX)p ∈ F
p
for all
t ∈ R. This implies that F
p
is totally geodesic at p.Since
˜
G acts transitively on
F
p
, F
p
is totally geodesic everywhere, which is a contradiction. So G cannot be
non-semisimple.
Remark 9.6.3 Let M = G/K be a Riemannian symmetric space with correspond-
ing Cartan decomposition g = k ⊕p. Then there is a one-to-one correspondence be-
tween the canonical G-invariant connections on M and the canonical G
∗
-invariant
connections on the dual Riemannian symmetric space M
∗
= G
∗
/K.Infact,assume
that M admits a canonical connection ∇
c
associated with a reductive decomposition
g = k ⊕m.Letg
∗
= k ⊕ip be the Lie algebra of G
∗
, regarded as a subspace of the
complexification g(C) of g.Letm
∗
be the subspace of g
∗
induced by m, via the nat-
ural vector spaces isomorphism g g
∗
.Thenm
∗
is an Ad
G
∗
(K)-invariant subspace
such that the geodesics through p = eK ∈M
∗
are given by one-parameter subgroups
with initial values in m
∗
.So∇
c
corresponds to a unique canonical connection on M
∗
.
Remark 9.6.4 Let us consider the sphere S
n
= SO
n+1
/SO
n
. Then, for all n = 3, the
Cartan decomp osition is th e only reductive decomposition associated with the pair
(SO
n+1
,SO
n
).Infact,if∇
c
is another canonical connection on S
n
, then the difference
tensor D = ∇ −∇
c
must be SO
n
-invariant. Let us assume first that n > 3. Let x,y,z ∈
T
e
1
S
n
and let V be the linear span of {x,y,z}. There exists g ∈SO
n
such that gu = −u
for all u ∈ V.ThenD
x
y,z = D
gx
gy,gz= −D
x
y,z and hence D
x
y,z = 0, which
implies D = 0 . Thus the Cartan decomposition is the unique reductive decom position
of the pair (SO
n+1
,SO
n
).Forn = 2 any 3-form is identically zero a nd so the assertion
is also true.
In contrast to n = 3, the 3-dimensional sphere S
3
is a Lie group and so there is
a one-parameter family of different canonical connections associated with the pair
(SO
4
,SO
3
) (Spin
3
×Spin
3
)/diag(Spin
3
×Spin
3
).
Remark 9.6.5 Let G =(G ×G)/diag(G ×G) be a simple Lie group of rank at least
2 and with a bi-invariant Riemannian metric. The difference Θ between any two
naturally reductive canonical connections is a totally skew 1-form with values in
the full isotropy algebra diag(g ×g) g. By identifying T
e
G with g and G with
diag(G ×G) we get that [g, Θ,G] is a symmetric irreducible and nontransitive skew-
torsion holonomy system.
Such a Θ is unique up to scalar multiple due to Proposition 9.3.3 (iv). So there ex-
ists a scalar
λ
such that Θ =
λ
ˆ
Θ,where
ˆ
Θ
v
·=
1
2
[v,·] is the difference tensor between
the Levi-Civita connection and the canonical connection ∇
c
1
associated with the re-
ductive complement m = g ×{0}⊂g ×g. (Recall, using the notation of Section 9.2,
that [ ¯v, ¯w]=−[v,w]; see Section A.4). This shows that the naturally reductive canon-
ical connections on G determine a line in the space of all connections, namely the
The Skew-Torsion Holonomy Theorem 257
affine line
{∇
c
t
=(1 −t)∇ + t∇
c
1
: t ∈ R}.
In Exercise 9.7.3 we present an explicit description of this line of connections in
terms of reductive decompositions.
Observe that any naturally reductive canonical connection o n G is invariant under
I
o
(G)=G ×G (effectivized). On the other hand, if
σ
is the geodesic symmetry at e,
then its induced action on covariant derivatives maps ∇
c
t
to ∇
c
−t
.So,
I
o
(G)=Aff
o
(G,∇
c
t
),
where Aff(G,∇
c
t
) are the isometries of G preserving ∇
c
t
(see Exercise 9.7.2). How-
ever,
I(G)=Aff(G,∇
c
t
)
holds if and only if the canonical connection coincides with the Levi-Civita connec-
tion.
Remark 9.6.6 Agricola, Ferreira, and Friedrich [6] classified recently the naturally
reductive spaces of dimension at most 6. Their approach has some relation with skew-
torsion holonomy systems. This is also the case o f [8], where Agricola and Friedrich
gave a conceptual proof of a well known classification by Cartan of connections with
skew torsion.
9.6.1 The full isometry group of naturally reductive spaces
The uniqueness of the canonical connection h as interesting applications to the
computation of the (full) group of isometries of a naturally reductive space.
Let M = G/H be an irreducible, simply connected, naturally reductive Riemann-
ian homogeneous space, where G is a connected subgroup of the full isometry group
I(M).Letg = h ⊕m be the naturally reductive decomposition an d ∇
c
be the asso-
ciated canonical connection on M.IfM = G/K is a Riemannian symmetric space,
where (G,K) is a symmetric pair, then I
o
(M)=G. So in o rder to compute I
o
(M)
we may assume that M is not a symmetric space and so M is neither isometric to a
sphere nor to a c ompact simple Lie group with a bi-invariant Riemannian metric, or
its symmetric dual.
Let Aff(M,∇
c
) denote the group of isometries of M preserving the canonical
connection ∇
c
.IfM is compact, then Aff
o
(M,∇
c
) coincides with the identity com-
ponent of the Lie group of ∇
c
-affine diffeomorphisms of M (see Exercise 9.7.2).
Any g ∈ I(M) maps the canonical connection ∇
c
into another canonical connection.
Theorem 9.6.1 then implies
Aff(M,∇
c
)=I(M).
Let us assume that G is the group of transvections Tr(∇
c
) of the canonical connection
∇
c
. Otherwise, we replace G by the normal Lie subgroup associated with the ideal
[m,m]+m of g. If the metric on M is G-normal homogeneous, that is, if m = h
⊥
258 Submanifolds and Holonomy
with respect to a bi-invariant Riemannian m etric on G,thenG = Tr(∇
c
). In fact, the
complementary ideal of [m, m]+m lies in h and so it is trivial since H acts effectively.
Recall that the group of transvections, which is a normal subgroup of Aff(M,∇
c
),
can be characterized as follows: an isometry g of M is a ∇
c
-transvection if for every
p ∈M there exists a piecewise smooth curve
γ
: [0,1] →M with
γ
(0)=p and
γ
(1)=
gp such that the ∇
c
-parallel transport along
γ
from p to gp coincides with d
p
g.Let
us assume, furthermore, that M is compact, Then I(M) is compact as well. Let b be
a complementary ideal of g in Lie(I(M)) = Lie(Aff(M,∇
c
)).Ifv ∈ b,then[v,g]=0
and so the associated Killing vector field ¯v on M is a G-invariant vector field on M.
Conversely, by Remark 9.2.1, any G-invariant vector field ˜u on M is a Killing vector
field. We write ˜u = ¯v + ¯w with v ∈ b and w ∈ g . Observe that w must belong to the
center z of g.Conversely,ifz ∈ z,then¯z is a G-invariant vector field on M. Then the
Lie algebra s of G-invariant Killing vector fields on M can be written as
s = b ⊕z.
If S is the connected component containing [e]=eH ∈ M of the set of fixed points of
the isotropy group H,thenS is a Lie group in a natural way:
S {g |
S
: gS = S}.
The Lie algebra o f S is naturally id entified with s by restricting the G-invariant
Killing vector fields on M to S.
So we have the following result, proved in [284] for normal homogeneous spaces
and in [262] for naturally reductive spaces.
Theorem 9.6.7 (Reggiani) Let M = G/H b e a compact, irreducible, simply con-
nected, naturally reductive Riemannian homogeneous space, where G = Tr(∇
c
) is
the transvection group of the canonical connection (this is always satisfied if the
metric on M is normal homogeneous with respect to G). Assume that M is not iso-
metric to a sphere or to a compact Lie group with a bi-invariant Riemannian metric.
Then
I
o
(M)=G
s
×K,
where G
s
is the semisimple part of G and K is the Lie group associated with the
Lie algebra of G-invariant vector fields on M (or, equivalently, K is the Lie group
naturally identified with the connected component containing [e] ∈ Mofthefixed
point set of H).
The above theorem is not true if the naturally reductive space is not presented by
means of the transvection group, as wrongly stated in [262]. In fact, if the Lie group
K is neither trivial nor abelian, we can choose a proper nontrivial subgroup K
⊂ K
that is not a normal subgroup of K.ThenM =(G
s
×K
)/(G
s
×K
)
[e]
is naturally
reductive with respect to the same naturally reductive complement m,butG
s
×K
is not a factor of I
o
(M)=G
s
×K as it would follow if we could apply the above
theorem, replacing G by G
s
×K
.
The Skew-Torsion Holonomy Theorem 259
Observe from Exercise 9.7.5 that Th eorem 9 .6.7 is also valid if M is isometric to
a Lie group with a bi-invariant Riemannian metric.
From the classification o f compact isotropy irreducible spaces [338, 346] it fol-
lows that the presentation group of any of such space, which is neither isometric to
a sphere nor to a Lie group, coincides with th e identity component of the isometry
group. We have the following corollary, which explains this fact conceptually. This
question was posed by Wolf for strongly isotropy irreducible spaces and by Wang
and Z iller in general.
Corollary 9.6.8 (Olmos, Reggiani [261]) Let M = G /H be a compact, simply con-
nected, irreducible, homogeneous Riemannian manifold that is not isometric to a
sphere or to a (simple) compact Lie group with a bi-invariant Riemannian metric.
If M is isotropy irreducible with respect to the pair (G,H) (effective action; G not
necessarily connected), then G
o
= I
o
(M).
Proof Observe first that G
o
is semisimple. In fact, let D be the distribution on M
given by the tangent spaces to the orbits of the maximal abelian normal (connected)
subgroup A of G
o
and assume that A = {e}. Such a distribution must be G-invariant
and so, since the action is effective, D
q
= T
q
M for some and hence for all q ∈ M.
Then A acts transitively on M and so M is flat, which is a contradiction.
Let us endow M with a normal homogeneous metric ·,·
with respect to the
decomposition g = h ⊕h
⊥
, where the inner product on T
p
M h
⊥
is the restriction
of −B and B is the Killing form of g.
The metric ·,·
must also be G-invariant, since any element of H preserves
both h and B.SinceH acts irreducibly on the tangent space, ·,·
coincides with the
Riemannian metric ·, · on M up to homothety. Since M is G
o
-normal homogeneous,
G
o
coincides with the group of transvections Tr(∇
c
) of the canonical connection.
Note that M = G
o
/H
o
since M is simply connected,.
From Theorem 9.6.7 we know that I
o
(M)=G
o
×K,whereK is the Lie group
whose Lie algebra consists of the G
o
-invariant vector fields on M. We shall prove
that K is trivial. Let V ⊂ T
p
M be the subspace of fixed vectors of H
o
. Observe that
the evaluation at p =[e] is a bijection between the set of G
o
-invariant vector fields
on M and V. The subspace V is H-invariant and thus V = {0} or V = T
p
M.Letus
assume that V = T
p
M.ThenK acts transitively on M. Moreover, the isotropy group
of K is trivial. In fact, dimK = dimM, which implies that the isotropy group must
be finite. Moreover, since M is simply connected, the isotropy group of K must be
trivial. On the other hand, H
o
is trivial, since it acts trivially on T
p
M = V.ThenG
o
acts, as well as K, simply transitively on M.SinceK commutes with G
o
we obtain
that K G
o
and M =(G ×G)/diag(G ×G) is isometric to a compact Lie group with
a bi-invariant Riemannian metric (see Exercise 9.7.4). This is a contradiction and
hence we must have V = {0} and so K is trivial.
9.6.2 The holonomy of naturally reductive spaces
Let M = G/H be an n-dim ensional, simply con nected, irreducib le, n aturally re-
ductive space with associated na turally reductive decomposition g = h ⊕m at p =[e]
260 Submanifolds and Holonomy
and canonical connection ∇
c
.LetD = ∇ −∇
c
, which is to tally skew. From Sec-
tion 9.2 we know that −D
v
w = D
w
v = ∇
v
¯w for all v,w ∈ T
p
M,andsoD
w
= ∇ ¯w.
By [9, 188], D
w
belongs to the (restricted) holonomy algebra for all w ∈ T
p
M
(see [92] for a proof). If M is not a symmetric space, then D = 0 and the holon-
omy group is transitive on the unit sphere by the Berger Holonomy Theorem 8.3.1.
Then [T
p
M,D,Hol
o
p
(M)] is a transitive skew-torsion holonomy system and therefore
Hol
o
p
(M)=SO
n
. Hence any non-symmetric naturally reductive space has generic
restricted holonomy. This result extends that of Wolf [346] for strongly isotropy irre-
ducible homogeneous spaces.
9.6.3 Spaces with the same isotropy as a group-type symmetric space
For the sake of completeness we include here a classification free proof of the
following result by Wolf (see [261], Proposition 8.1).
Proposition 9.6.9 (Wolf [346]) Let M = G/H be a simply connected, compact, ho-
mogeneous Riemannian manifold such that H acts on T
p
M, p =[e], as the adjoint
representation of a compact semisimple Lie group. Then M is isometric to a compact
semisimple Lie group with a bi-invariant Riemannian metric.
Proof If the rank k of the compact semisimple Lie group, which coincides with the
codimension of the principal H-orbits, is equal to 1, then M has dimension 3 and
H = SO
3
by Remark 9.3.4. Hence M is isometric to the sphere S
3
and the conclusion
holds. Thus we assume k ≥ 2. By assumption we can identify h with T
p
M. Then, via
the isotropy representation, H acts as the adjoint representation of H on h T
p
M.
For 0 = v ∈ h the normal space of the orbit H ·v at v is
ν
v
(H ·v)=C (v)={
ξ
∈h : [
ξ
,v]=0}.
We have v = Exp(tv)v = Ad(Exp(tv))(v),andsoAd(Exp(tv)) leaves
ν
v
(H ·v) in-
variant for all t ∈ R. Mo reover, from the above equality, the set of fixed points of
the one-parameter group of linear isometries {Ad(Exp(tv)) : t ∈ R} of h is just the
normal space
ν
v
(H ·v) (see Remark 9.4.3). Let us now write
d
p
h
t
= Ad(Exp(tv))
with h
t
∈ H.ThenS = {h
t
: t ∈ R} is a one-parameter group of isometries such that
M
v
= exp
p
(
ν
v
(H ·v)) is the connected component containing p of the set of fixed
points of S. Clearly, M
v
is a totally geodesic submanifold of M. Observe that M
v
is
a homogeneous submanifold of M by Lemma 9.1.1. Moreover, it is not hard to see
that the isotropy algebra of M
v
is C (v), since this algebra coincides with its own
normalizer in h. In the case that H ·v, v = 0, is a most singular orbit (that is, v is in
a one-dimensional simplex of the Weyl chamber in a maximal abelian subspace, or
equivalently, the abelian part of the centralizer of v in h is one-dimensional) we have
C (v)=Rv ⊕
˜
h
v
,
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