324 Submanifolds and Holonomy
12.2 Polar actions — higher rank
The first complete classification of polar actions on a Riemannian symmetric
space of compact type and higher rank was obtained by Podest`a and Thorbergs-
son in [278]. They classified explicitly all connected compact subgroups of SO
n+2
,
n ≥ 3, acting polarly on the complex quadric Q
n
= SO
n+2
/SO
n
SO
2
. A remarkable
consequence of this classification is that each polar action on the complex quadric
Q
n
= SO
n+2
/SO
n
SO
2
, n ≥ 3, is hyperpolar.
Biliotti and Gori then obtain ed in [45] the classification of all connected com-
pact subgroups of SU
n
acting polarly on the complex Grassmannian G
m
(C
n
)=
SU
n
/S(U
m
U
n−m
),2≤m ≤
n
2
. Again, each of these actions is hyperpolar.
The complex quadrics Q
n
and the complex Grassmannians G
m
(C
n
) are examples
of Hermitian symmetric spaces. Biliotti exten ded in [44] the work by Podest`aand
Thorbergsson and by Biliotti and Gori and obtained the classification of all polar
actions on irreducible Hermitian symmetric sp aces of compact type. From Biliotti’s
work it followed that each polar action on an irreducible Hermitian symmetric space
of compact type is hyperpolar. Biliotti conjectured in [44] that a polar action on a
compact symmetric space of rank bigger than one is hyperpolar.
In the above three papers the authors acutally considered a larger class of actions,
namely coisotropic actions. An action of a compact subgroup of a compact K¨ahler
manifold M is coisotropic if the principal orbits of the action are coisotropic sub-
manifolds of M with respect to its K¨ahler form. Podest`a and Thorbergsson proved
in [278] that every polar action on an irreducible compact homogeneous K¨ahler ma-
nifold is coisotropic. This approach works well for Hermitian symmetric spaces of
compact type.
Kollross gave in [181] an affirmative answer to Biliotti’s conjecture for compact
symmetric spaces of rank greater than one whose isometry group is simple, that is,
for irreducible Riemannian symmetric spaces of Type I and rank greater than one. For
the compact exceptional simple Lie groups, Kollross proved the conjecture in [182].
This just left the conjecture open for the compact classical simple Lie groups.
With a completely d ifferent approach, Lytchak proved in [202] that each polar
action with cohomogeneity ≥ 3 on a simply connected Riemannian symmetric space
of compact type is hyperpolar. In fact, Lytchak proved a similar statement for the
more general class of polar foliations.
The remaining case of cohomogeneity two on the compact classical simple Lie
groups was settled jointly by Kollross and Lytchak in [185]. The combination of the
above work leads to
Theorem 12.2.1 Any polar action on an irreducible Riemannian symmetric space of
compact type and with rank ≥2 that has an orbit of positive dimension is hyperpolar.
The special case that the polar action has a fixed point was already investigated
by Br¨uck in her thesis [53]. In this situation the action is orbit equivalent to the action
Polar Actions on Symmetric Spaces of Compact Type 325
of the connected isotropy group of the isometry group of the symmetric space (see
also [181]).
12.3 Hyperpolar actions — higher rank
The first thorough investigation of hyperpolar actions on symmetric spaces is in
the work [148] by Heintze, Palais, Terng, and Thorbergsson. In [149] they developed
a structure theory for hyperpolar actions with a fixed point on compact homogeneous
Riemannian manifolds.
There are two important types of examples of hyperpolar actions on symmetric
spaces:
1. Cohomogeneity one actions [148]: The principal orbits of the action have co-
dimension one. The main point here is that the geodesics intersecting the or-
bits perpendicularly are closed. For this the assumption that th e action is by
a closed subgroup of the isometry group is essential. For a counterexample in
the non-closed case see Remark 5.6.8 in [275].
2. Hermann actions [153]: Let (G,K) be a Riemannian symmetric pair with a
compact Lie group G and let H be a subgroup of G such that h is the fixed
point set of an involution of g. Then the action of H on the Riemannian sym-
metric space G/K is hyperpolar (or equivalently, the action of H ×K on G is
hyperpolar)
Kollross proved in [180] that, up to orbit equivalence, there are no other hyper-
polar actions on irreducible Riemannian symmetric spaces of compact type:
Theorem 12.3.1 (Kollross) Let M = G/K be an irreducible Riemannian symmetric
space of compact type. Then every hyperpolar action on M is either of cohomogeneity
one or orbit equivalent to a Hermann action.
Tables containing Hermann actions and orbit equivalent subactions can be found
in [181]. We shall now give a proof of the original result by Hermann [153]:
Theorem 12.3.2 (Hermann) Let (G,K) and (G,H) be two Riemannian symmetric
pairs. Then the action of H on the Riemannian symmetric space M = G/K is hyper-
polar.
Proof By Exercise 12.6.2 it suffices to show that the action of H ×K on G is hyper-
polar. The left mu ltiplication L
g
in G maps g
−1
Hg ×K-orbits onto H ×K-orbits. It
follows that HgK is a principal H ×K-orbit if and only if e is a principal g
−1
Hg×K-
orbit. The H ×K-action on G is hyperpolar if and only if the g
−1
Hg ×K-action on
G is hyperpolar. So, by replacing H by g
−1
Hg ×K, we may assume that the identity
326 Submanifolds and Holonomy
e ∈ G lies on a principal H ×K-orbit, that is, HK is a principal orbit of the H ×K-
action on G.Let
g = k ⊕p and g = h ⊕p
be the Cartan decompositions associated with the Riemannian symmetric pairs
(G,K) and (G, H), respectively. Then the normal space
ν
e
(HK) at e of the orbit
(H ×K) ·e = HK is given by
ν
e
(HK)=p
∩p.
The isotropy group of H ×K at e is given by
(H ×K)
e
= {(k,k) : k ∈H ∩K}.
Since the orbit (H ×K) ·e is princip al, (H ×K)
e
acts trivially on the normal space
ν
e
(HK).Thus,wehaveAd(k)v =(k,k)v = v for all k ∈ H ∩K and v ∈ p ∩p
,which
implies that
[h ∩k,p
∩p]={0}.
Since
[
ν
e
(HK),
ν
e
(HK)] = [p
∩p,p
∩p] ⊂ h ∩k,
we conclude that the normal space
ν
e
(HK) is an abelian subspace of g. Moreover,
using Exercise 12.6.3, we see that Σ = exp
e
(
ν
e
(HK)) = Exp(
ν
e
(HK)) is a compact
flat totally geodesic submanifold of G. Finally, using Exercises 12.6.4 and 12.6.5, we
conclude that Σ is a section of the (H ×K)-action on G and therefore the action of
H ×K on G is hyperpolar.
12.4 Cohomogeneity one actions — higher rank
Some classifications of cohomogeneity one actions were obtained by Uchida,
Iwata and Kuroki by using cohomological methods. Uchida investigated in [330]
cohomogeneity one actions by compact Lie groups on simply connected smooth
manifolds whose rational cohomology ring H
∗
(M,Q) is isomorphic to the rational
cohomology ring of a complex projective space. This includes for example the odd-
dimensional complex quadrics Q
2k+1
= SO
2k+3
/SO
2k+1
SO
2
, k ≥ 1. Uchida proved
that, up to isomorphism, the only cohomogeneity one actions on odd-dimensional
complex quadrics are the actions of SO
2k+2
⊂ SO
2k+3
on Q
2k+1
and of Spin
7
⊂
SO
8
⊂ SO
9
on Q
7
.
Kuroki investigated in [192] cohomogeneity one actions by compact Lie groups
on simply connected smooth manifolds whose rational cohomology ring is iso-
morphic to the rational cohomology ring of an even-dimensional complex quadric
Q
2k
= SO
2k+2
/SO
2k
SO
2
, k ≥ 2. Up to isomorphism, the only such actions are the
Polar Actions on Symmetric Spaces of Compact Type 327
standard actions of SO
2k+1
,U
k+1
,SU
k+1
⊂SO
2k+2
on Q
2k
, Sp
k
Sp
1
⊂SO
4k
on Q
4k−2
,
Spin
9
⊂ SO
16
on Q
14
, G
2
⊂ SO
7
⊂ SO
8
on Q
6
, and a non-standard action of G
2
U
1
on Q
6
.
Iwata [161] investigated with similar methods the analogous problem for man-
ifolds whose rational cohomology ring is isomorphic to that of a quaternionic pro-
jective space. This includes the exceptional symmetric space G
2
/SO
4
.Iwatashowed
that, up to isomorphism, every cohomogeneity one action on the exceptional sym-
metric space G
2
/SO
4
is isomorphic to the standard action of SU
3
⊂ G
2
on G
2
/SO
4
.
Already the case of the complex quadrics shows that the cohomological methods
are in general less suitable for the classification problem of cohomogeneity one ac-
tions on Riemannian symmetric spaces of compact type. Since cohomogeneity one
actions are hyperpolar, the classification by Kollross [180] of hyperpolar actions in-
cludes the classification o f cohomogeneity one actions as a special case.
We describe the idea for the classification by Kollross in the special case when
the action is of cohomogeneity one and the symmetric space M = G/K is of rank
≥2 and not of group type. Suppose H is a maximal closed subgroup of G.IfH is not
transitive on M, then its cohomogeneity is at least one. Since the cohomogeneity of
the action of any closed subgroup of H is at least the cohomogeneity of the action of
H, and we are interested only in classification up the orbit equivalence, it is sufficient
to consider only maximal closed subgroups of G. But it may happen that H acts
transitively on G/K. This happens in precisely four cases, where we write down
G/K = H/(H ∩K):
SO
2n
/U
n
= SO
2n−1
/U
n−1
(n ≥ 4),
SU
2n
/Sp
n
= SU
2n−1
/Sp
n−1
(n ≥ 3),
G
+
2
(R
7
)=SO
7
/SO
5
SO
2
= G
2
/U
2
,
G
+
3
(R
8
)=SO
8
/SO
5
SO
3
= Spin
7
/SO
4
.
In these cases, one has to go one step further and consider maximal closed subgroups
of H that then never happen to act also transitively. Thus, it is sufficient to consider
maximal closed subgroups of G, with the few exceptions just mentioned. In order
for a closed subgroup H to act with cohomogeneity one, it obviously must satisfy
dimH ≥ dim M −1. This already rules out a lot of possibilities. For the rem aining
maximal closed subgroups, one has to calculate the cohomogeneity case by case. One
way to do this is to calculate the cohomogeneity of the slice representation; this is the
action of the isotropy group H ∩K on the normal space at the corresponding point of
the orbit through that point. This procedure eventually leads to the classification of all
cohomogeneity one actions up to orbit equivalence, and hence to the classification of
homogeneous hypersurfaces in M = G/K. It turns out that, with five exceptions, all
homogeneous hypersurfaces arise via the construction of Hermann. The exceptions
come from the following actions:
1. The action of G
2
⊂ SO
7
on SO
7
/U
3
= SO
8
/U
4
= G
+
2
(R
8
)=Q
6
.
2. The action of G
2
⊂ SO
7
on SO
7
/SO
4
SO
3
= G
+
3
(R
7
).
328 Submanifolds and Holonomy
3. The action of Spin
9
⊂ SO
16
on SO
16
/SO
14
SO
2
= G
+
2
(R
16
)=Q
14
.
4. The action of Sp
n
Sp
1
⊂ SO
4n
on SO
4n
/SO
4n−2
SO
2
= G
+
2
(R
4n
)=Q
4n−2
.
5. The action of SU
3
⊂ G
2
on G
2
/SO
4
.
All other homogeneous hypersurfaces can be obtained via the construction of
Hermann. We refer to [180] for an explicit list of all Hermann actions of cohomo-
geneity one.
12.5 Hypersurfaces with constant principal curvatures
We first introduce two notions. A hypersurface M of a Riemannian manifold
¯
M is called curvature-adapted if its shape operator and its normal Jacobi operator
commute with each other. Recall that the normal Jacobi operator of M is the self-
adjoint (local) tensor field on M defined by
¯
R(.,
ξ
)
ξ
,where
¯
R is the Riemannian
curvature tensor of
¯
M and
ξ
is a (local) unit normal vector field on M.If
¯
M is a space
of constant cur vature, then the normal Jaco bi operator is a mu ltiple of the identity at
each point, and hence every hypersurface is curvature-adapted. However, for more
general ambient spaces this condition is quite restrictive. For instance, in a nonflat
complex space form, say CP
n
or CH
n
, a hypersurface M is curvature-adapted if and
only if the structure vector field on M is a principal curvature vector everywhere.
Recall that the structure vector field of M is the vector field obtained by rotating
a local unit normal vector field to a tangent vector field using the am bient K¨ahler
structure.
A (real) hypersurface M in a K¨ahler manifold
¯
M is called a Hopf hypersurface if
J(
ν
M) is invariant under the shape operator of M,whereJ is the complex structure of
¯
M and
ν
M is the normal bundle of M. From the explicit expression of the curvature
tensor of CP
n
(or CH
n
) one can easily deduce that a real hypersurface in CP
n
(or
CH
n
) is a Hopf hypersurface if and only if it is curvature-adapted.
We denote by g the number of d istinct principal curvatures of a hypersurface with
constant principal curvatures.
12.5.1 ... in complex projective s paces
The classification of hypersurfaces with constant principal curvatures in real
space forms was discussed in Section 2.9. We saw in Theorem 2.9.3 that an isopara-
metric hypersurface in a real space form has constant principal curvatures. This does
not hold in general in other Riemannian manifolds. For example, Wang [339] gave
a nice criterion for an isoparametric hypersurface in CP
n
to have constant principal
curvatures.
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