Polar Actions on Symmetric Spaces of Compact Type 329
Theorem 12.5.1 (Wang) Let M be an isoparametric hypersurface in CP
n
. Then the
following statements are equivalent:
(1) M has constant principal curvatures;
(2) M has a complex focal variety;
(3) M is a Hopf hypersurface;
(4) M is curvature-adapted.
In [339] Wang showed that none of two focal varieties of the inhomogeneous
isoparametric hypersurfaces in CP
n
that were constructed by Ozeki and Takeuchi
[272, 273] has a complex focal variety. It follows therefore from the above criterion
that these inhomogeneous isoparametric hypersurfaces do not have constant principal
curvatures. Wang’s result provides a justification for investigating hypersurfaces with
constant principal curvatures and isoparametric hypersurfaces separately.
Tashiro and Tachibana [315] proved that there are no totally umbilical hypersur-
faces in non flat complex space forms. Thus, we have g ≥ 2 for any hypersurface with
constant principal curvatures in CP
n
(and CH
n
). For CP
n
the classifications with
g ∈{2,3} were obtained by Takagi ( [307] for g = 2; [308] for g = 3andn ≥ 3) and
Wang ( [340] for g = 3andn = 2).
Theorem 12.5.2 (Takagi) Let M be a connected real hypersurface in CP
n
,n≥ 2 ,
with two distinct constant principal curvatures. Then M is an open part of a geodesic
hypersphere in CP
n
.
Note that a geodesic hypersphere in CP
n
is a tube around a totally geodesic CP
0
(a point) in CP
n
. Any geodesic hypersphere has two focal sets, a point and a totally
geodesic hyperplane CP
n−1
. Thus a geodesic hypersphere can also be considered as
a tube around a totally geodesic CP
n−1
. Cecil and Ryan [71] improved the above
result for n ≥ 3 by requiring that M has at most two distinct principal curvatures at
each point.
For three distinct constant principal curvatures we have:
Theorem 12.5.3 (Takagi (n ≥ 3); Wa ng (n = 2)) Let M be a connected real hyper-
surface in CP
n
,n≥ 2, with three distinct constant principal curvatures. Then M is
an open part of
(1) a tube around a k -dimensional totally geodesic subspace CP
k
⊂CP
n
for some
k ∈{1,...,n −2},or
(2) a tube around the complex quadric Q
n−1
⊂ CP
n
.
The focal set of a totally geodesic CP
k
in CP
n
is a totally geodesic CP
n−k−1
,and
the focal set of the complex quadric in CP
n
is a totally geodesic real projective space
RP
n
⊂ CP
n
. Thus, a tube around CP
k
is a tube around CP
n−k−1
, and a tube around
the complex quadric can be considered as a tube around RP
n
.
330 Submanifolds and Holonomy
It is clear that every homogeneous hypersurface has constant principal curvatures.
The homogeneous hypersurfaces in CP
n
were classified by Takagi (see Theorem
12.1.2). It follows from Takagi’s classification that every homogeneous hypersurface
in CP
n
is a Hopf hypersurface. Kimura [169] proved a kind of converse statement.
Theorem 12.5.4 (Kimura) Every connected Hopf hypersurface M in CP
n
,n≥ 2,
with constant principal curvatures is an open part of a homogeneous real hypersur-
face in CP
n
.
The proof goes roughly like this. The inverse image of M under the Hopf
map S
2n+1
→ CP
n
is a hypersurface in S
2n+1
with constant principal curvatures.
M¨unzner’s [222,223] result on the number of distinct principal curvatures of isopara-
metric hypersurfaces in spheres implies that g ∈{2,3,5}. The cases g ∈{2,3}fo llow
from the above classifications and only the case g = 5 remains to be analyzed. The
Hopf condition implies that M (locally) lies o n a tube around a complex submani-
fold. Kimura proved that this complex submanifold has parallel second fundamental
form and therefore is a symmetric complex submanifold of CP
n
. Using Nakagawa
and Takagi’s classification of such submanifolds (see Theorem 11.3.5), Kimura then
deduced his result.
There is still the following open problem:
O
PEN PROBLEM: Does there exist a connected real hypersurface with constant
principal curvatures in CP
n
which is not an open part of a homogeneous hypersurface
in CP
n
?
Any such hypersurface cannot be a Hopf hypersurface in view of Kimura’s re-
sult. The standard approach would be to investigate thoroughly the Gauss-Codazzi
equations, but the complexity of the Gauss-Codazzi equations for non-Hopf hyper-
surfaces in CP
n
makesthisadifficult problem. One possible approach was pursued
by D´ıaz-Ramos and Dom´ınguez-V´azquez in [99]. Denote by d the number of non-
trivial orthogonal projections from the principal curvature spaces onto the rank one
subbundle J(
ν
M) of TM. Thus M is a Hopf hypersurface if and only if d = 1. D´ıaz-
Ramos and Dom´ınguez-V´azquez proved that there are no real hypersurfaces with
constant principal curvatures in CP
n
with d = 2.
12.5.2 ... in quaternionic projective spaces
The first systematic study of real hypersurfaces with constant principal curvatures
in HP
n
was done by Mart´ınez and P´erez [204]. Let M be a real hypersurface in HP
n
.
The quaternionic K¨ahler structure induces a maximal quaternionic subbundle D of
the tangent bundle TM.Mart´ınez and P´erez imposed th e condition that D is invariant
under the shape operator of M, which is equivalent for M to be curvature-adapted.
They classified all curvature-adapted real hypersurfaces with constant principal cur-
vatures in HP
n
, and it follows from their classification that any such hypersurface
is an open part of a homogeneous hypersurface. For the classification of homoge-
neous hypersurfaces in HP
n
see Theorem 12.1.4. Thus, Mart´ınez and P´erez obtained
Polar Actions on Symmetric Spaces of Compact Type 331
the analogue for HP
n
of Kimura’s Theorem 12.5.4. Surprisingly, as was proved by
Berndt in [21], every curvature-adapted real hypersurface in HP
n
has constant princi-
pal curvatures. One can easily deduce from the classification that every homogeneous
hypersurface in HP
n
is curvature-adapted. We therefore have the following charac-
terization of homogeneous hypersurfaces in HP
n
in ter ms of an algebraic curvature
condition:
Theorem 12.5.5 (Berndt) A connected real hypersurface M in HP
n
,n≥ 2,is
curvature-adapted if and only if M is an open part of a homogeneous real hyper-
surface in HP
n
.
Also here we have the open problem:
O
PEN PROBLEM: Does there exist a connected real hypersurface with constant
principal curvatures in HP
n
which is not an open part of a homogeneous hypersurface
in HP
n
?
Any such hypersurface cannot be a curvature-adapted in view of the previous
result. Mart´ınez and P´erez showed in [204] that every real hypersurface in HP
n
with
at most two distinct principal curvatures at each point is an open part o f a geodesic
hypersphere in HP
n
,butforg ≥ 3noclassification results are k nown.
12.5.3 ... in Cayley projective plane
For the Cayley projective plane OP
2
an analogue was proved by Murphy [224].
Theorem 12.5.6 (Murphy) Let M be a connected curvature-adapted real hypersur-
face in OP
2
. Then M has constant principal curvatures if and only if M is an open
part of a homogeneous real hypersurface in OP
2
.
For the classification of homogeneous hypersurfaces in OP
2
see Theorem 12.1.5.
12.6 Exercises
Exercise 12.6.1 Let M = G/K be a compact Hermitian symmetric space with G =
I
o
(M), r = rk(M) and n + 1 = dim
C
M. Prove that the induced action by K on CP
n
is
polar and the sections are totally geodesic RP
r−1
⊂ CP
n
.
Exercise 12.6.2 Let (G,K) be a Riemannian symmetric pair such that M = G/K
is a compact simply connected Riemannian symmetric space. Endow G with the
corresponding bi-invariant Riemannian metric and let H be a Lie subgroup of G.
332 Submanifolds and Holonomy
Prove that the action of H on M is hyperpolar if and only if the action of H ×K on G
given by
(H ×K) ×G → G , ((h,k),g) → hgk
−1
is hyperpolar.
Exercise 12.6.3 Let (G,K) be a Riemannian symmetric pair such th at M = G/K is a
compact Riemannian symmetric space. Endow G with the corresponding bi-invariant
Riemannian metric and let g = k ⊕p be a Cartan deco mposition of g and Exp : g →G
be the Lie exponential map. Prove
(a) Σ = Exp(p) is a compact totally geodesic submanifold of G.
[Hint: Let
σ
be the involution of G associated with (G, K) and let
τ
be the sym-
metry of G at e,i.e.
τ
(g)=g
−1
.ThenΣ is the connected component containing
e of the fixed point set of the isometry
τ
◦
σ
of G.]
(b) Consider the Cartan decomposition g ⊕g =
˜
k ⊕
˜
p of the Lie algebra of G =
(G ×G)/diag(G ×G),where
˜
k = {(X,X ) : X ∈ g},
˜
p = {(X,−X) : X ∈ g}.
Then T
p
Σ is associated with the Lie triple system {(X ,−X) : X ∈ p} of
˜
p.
Exercise 12.6.4 (Lemma of Hermann). Let F be a flat compact submanifold of a
Riemannian manifold M and let X be a Killing vector field on M.Let
¯
X be the
Killing vector field on F that is given by projecting X
q
orthogonally onto T
q
F for all
q ∈ F. Prove:
(a)
¯
X is a parallel vector field on F.
[Hint: Use the facts that
¯
X is bounded and F is flat.]
(b) If
¯
X
q
= 0forsomeq ∈ F,then
¯
X = 0.
Exercise 12.6.5 Let M be a connected Riemannian manifold and let G be a compact
subgroup of I(M). Show that for every p ∈M the image exp
p
(
ν
p
(G·p)) of the normal
space
ν
p
(G · p) of the orbit G · p at p under the exponential map exp
p
: T
p
M → M
intersects all G-orbits in M. [Hint: Any geodesic
γ
: [0,1] → M with
γ
(0)=p and
q =
γ
(1) /∈ G · p that minimizes the distance from q to the compact set G · p must be
perpendicular at t = 0 to the tangent space T
γ
(0)
(G ·p).]
Exercise 12.6.6 Prove that the focal set of the complex quadric Q
n−1
⊂ CP
n
is a
totally geodesic RP
n
⊂ CP
n
.
Exercise 12.6.7 The action of Sp
3
Sp
1
⊂ F
4
on the Cayley p rojective plane OP
2
has
a totally geodesic HP
2
as a singular orbit. Prove that the second singular orbit is an
11-dimensional sphere S
11
.
Exercise 12.6.8 Show that the action of the exceptional Lie group G
2
⊂ SO
7
on
SO
7
/U
3
= SO
8
/U
4
= G
+
2
(R
8
)=Q
6
is of cohomogeneity one. Prove that the two
singular orbits of this action are G
2
/U
2
= G
+
2
(R
7
)=Q
5
and G
2
/SU
3
= S
6
.
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