358 Submanifolds and Holonomy
by Cartan. There are two symmetric spaces of noncompact type with a triality prin-
ciple, namely
SO
8
(C)/SO
8
and SO
o
4,4
/SO
4
SO
4
.
Each symmetry P ∈ A extends canonically to a linear isometry of a
∗
and, via the
inner product given by
α
,
β
= H
α
,H
β
, to a linear isometry of a.Sincea is an
r-dimensional real vector space, we get a canonical action of A on the real projective
space RP
r−1
induced by a.
Now consider the Iwasawa decompositions g = k ⊕a ⊕n of g and G = KAN
of G. The solvable Lie group AN acts simply transitively on M and therefore M is
isometric to the connected, simply connected, solvable Lie group AN equipped with
the left-invariant Riemannian metric that is induced from the inner product ·,·.Let
∈ RP
r−1
be a linear line in a.Since lies in the orthogonal complement of the
derived algebra n of a ⊕n, the orthogonal complement s
=(a ⊕n) of in a ⊕n
is a subalgebra of a ⊕n of codimension one. Let S
be the connected Lie subgroup of
AN with Lie algebra s
. Then the orbits of the action of S
on M form a Riemannian
foliation F
of M whose leaves are homogeneous hypersurfaces. If M has rank one,
then a is one-dimensional and hence there exists only one such foliation, namely the
one given by S
= S
a
= N. This is precisely the horosphere foliation of M,allof
whose leaves are isometrically congruent to each other.
Proposition 13.4.3 ( [37]) Let M be a Riemannian symmetric space of noncompact
type. The homogeneous codimension one foliation F
of M has the following proper-
ties:
(1) All leaves of F
are isometrically congruent to each other.
(2) Let H ∈ be a unit vector. Then the shape operator A
H
of the leaf of F
at o
with respect to H is
A
H
= ad(H) : s
→ s
.
In particular, the principal curvatures are 0 and
α
(H),
α
∈ Ψ
+
, with corre-
sponding principal curvature spaces a and g
α
, respectively.
(3) The mean curvature
μ
of each leaf of F
is
μ
=
1
dim(M) −1
∑
α
∈Ψ
+
dim(g
α
)
α
(H).
In particular, if r = rk(M ) > 1, then there exists ∈ RP
r−1
such that all leaves
of F
are minimal hypersurfaces and hence F
is a harmonic foliation. In this
case the projection from M onto the orbit space M/S
= R is a harmonic map.
Using structure theory of semisimple and solvable Lie algebras, one can show
that two foliations F
and F
are isometrically congruent to each other if and only if
there exists a symmetry P ∈A with P()=
. It follows that the set of all congruence
classes of such foliations is parametrized by RP
r−1
/A.
Polar Actions on Symmetric Spaces of Noncompact Type 359
Let
α
i
∈ Λ, i ∈{1,...,r}, be a simple root. For each unit vector
ξ
∈ g
α
i
the
subspace s
ξ
= a ⊕(n R
ξ
) is a subalgebra of a ⊕n.LetS
ξ
be the connected Lie
subgroup of AN with Lie algebra s
ξ
. Then the orbits of the action of S
ξ
on M form
a Riemannian f oliation F
ξ
of M whose leaves are homogeneous hypersurfaces. If
η
∈ g
α
i
is another unit vector, the induced foliation F
η
is congruent to F
ξ
under an
isometry in the centralizer K
0
of a in K. Thus, for each simple root
α
i
∈Λ, we obtain
a congruence class of homogeneous foliations of codimension one of M. We denote
a representative of this congruence class by F
i
. By investigating the geometry of
these foliations one can prove that F
i
and F
j
are isometrically congruent if and only
if there exists a symmetry P ∈ A with P(
α
i
)=
α
j
. Thus, the set of all congruence
classes of such foliations is param etrized by {1,...,r}/A, where the action of A on
{1,...,r} is given by identifying k ∈{1,...,r} with the vertex
α
k
in the Dynkin
diagram. The geometry of these foliations is quite fascinating. We p ut S
i
= S
ξ
and
denote by F
i
= S
i
·o the leaf of F
i
containing o..
Proposition 13.4.4 ( [37]) Let M be a Riemannian symmetric space of noncompact
type. The homogeneous codimension one foliation F
i
of M has the following proper-
ties:
(1) The leaf F
i
is the only minimal leaf in F
i
and the mean curvature
μ
(t) of the
leaf of F
i
at distance t > 0 fro m F
i
is
dimg
α
i
+ 2dimg
2
α
i
dimM −1
|
α
i
|tanh(|
α
i
|t).
(2) The two minimal leaves F
i
and F
j
are isometric to each other if and only if
there exists a symmetry P ∈ A with P(
α
i
)=
α
j
.
(3) If |
α
i
| = |
α
j
|, then corresponding leaves in F
i
and F
j
have the same principal
curvatures with the same multiplicities.
In item (3), by corresponding leaves we mean the two minimal leaves F
i
and F
j
,
and leaves at the same positive distance from the se two minimal leaves. All leaves of
F
i
together form a homogeneous isoparametric system on M, and if the rank of M is
≥ 3, there exist among these systems some that are noncongruent but have the same
principal curvatures with the same multiplicities. Such a feature had already been
discovered by Ferus, Karcher, and M¨unzner [131] for inhomogeneous isoparametric
systems on spheres.
Example 13.4.2 ( [37]) Consider the symmetric space M = SL
4
(R)/SO
4
of rank r =
3 and dimension 9. The Dynkin diagram is
α
1
α
2
α
3
and the nontrivial Dynkin diagram automorphism in A
2
is given by
α
1
→
α
3
,
α
2
→
α
2
and
α
3
→
α
1
. The homogeneous codimension one foliations F
1
and F
3
360 Submanifolds and Holonomy
are isometrically congruent to each other, whereas F
1
and F
2
are not isometrically
congruent to each other. We identify M with the solvable Lie subgroup in the usual
Iwasawa decomposition of SL
4
(R),thatis,
M = SL
4
(R)/SO
4
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
X =
⎛
⎜
⎜
⎝
x
11
x
12
x
13
x
14
0 x
22
x
23
x
24
00x
33
x
34
000x
44
⎞
⎟
⎟
⎠
: x
ij
∈ R, det(X)=1
⎫
⎪
⎪
⎬
⎪
⎪
⎭
.
The minimal leaves F
1
in F
1
and F
2
in F
2
are
F
1
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎛
⎜
⎜
⎝
x
11
0 x
13
x
14
0 x
22
x
23
x
24
00x
33
x
34
000x
44
⎞
⎟
⎟
⎠
⎫
⎪
⎪
⎬
⎪
⎪
⎭
, F
2
=
⎧
⎪
⎪
⎨
⎪
⎪
⎩
⎛
⎜
⎜
⎝
x
11
x
12
x
13
x
14
0 x
22
0 x
24
00x
33
x
34
000x
44
⎞
⎟
⎟
⎠
⎫
⎪
⎪
⎬
⎪
⎪
⎭
.
Both F
1
and F
2
are homogeneous hypersurfaces in M with the same principal curva-
tures and the same multiplicities for corresponding principal curvatures. Explicitly,
the principal curvatures are 0,
1
2
and −
1
2
with multiplicities 4, 2 and 2, respectively.
However, F
1
and F
2
are not (intrinsically) isometric to each other. Furthermore, the
homogeneous hypersurfaces in F
1
and F
2
at a fixed positive distance to F
1
and F
2
,
respectively, also have the same principal curvatures and the same multiplicities for
corresponding principal curvatures. It follows that one cannot use principal curva-
tures and their multiplicities as geometric invariants to distinguish homogeneous hy-
persurfaces in a Riemannian symmetric space of noncompact type. It is not known
whether in the compact case one can use principal curvatures as geometric invariants
for this purpose.
Using structure theory of semisimple and solvable Lie algebras, Berndt and
Tamaru proved in [37] that every homogeneous codimension one foliation of M is
isometrically congruent to one of the above foliations:
Theorem 13.4.5 (Berndt, Tamaru) Let M be a connected irreducible Riemannian
symmetric space of noncompact type and with rank r. The moduli space M
F
of all
noncongruent homogeneous codimension one foliations of M is isomorphic to the
orbit space of the action of A on RP
r−1
∪{1 ,...,r}:
M
F
∼
=
(RP
r−1
∪{1 ,...,r})/A .
Proof (Sketch) Let H be a connected closed subgroup of G = I
o
(M) which acts on
M with cohomogeneity one and has no singular orbits.
The first step is to show that there exists a connected solvable closed subgroup of
H which acts simply transitively on each orbit. This can be achieved by combining
a Levi decomposition of H with an Iwasawa decomposition of the semisimple factor
in the Levi decomposition and then applying Cartan’s Fixed Point Theorem. Then
we can assume, without loss of generality, that H has these properties.
The second step is to show that there exists an Iwasawa decomposition g = k ⊕
a ⊕n of g such that h = t ⊕a ⊕n,wheret is a m aximal abelian subalgebra of the
Polar Actions on Symmetric Spaces of Noncompact Type 361
centralizer k
0
of a in k. The projection h
a⊕n
of h onto a ⊕n with respect to this
decomposition is a subalgebra of a ⊕n. The codimension one subalgebras of a ⊕n
can be shown to be of the form (a ⊕n) ,where is a one-dimensional subspace of
a or of RH
α
⊕g
α
with some simple root
α
∈ Λ.If ⊂ a or ⊂ g
α
,thenH and the
connected closed subgroup H
a⊕n
of G with Lie algebr a h
a⊕n
give the same foliation .
In the other case one can show that H and H
a⊕n
are orbit equivalent.
It is surprising and remarkable that M
F
depends only on the rank and on possible
duality or triality principles on the symmetric space. For example, for the symmetric
spaces
SO
17
(C)/SO
17
, Sp
8
(R)/U
8
, Sp
8
(C)/Sp
8
, SO
∗
32
/U
16
, SO
∗
34
/U
17
,
E
8
8
/SO
16
, E
8
(C)/E
8
, SO
8,n+8
/SO
8
SO
n+8
(n ≥ 1),
SU
n,n+8
/S(U
8
U
n+8
)(n ≥ 0), Sp
n,n+8
/Sp
8
Sp
n+8
(n ≥ 0)
the set M
F
of all noncongruent homogeneous codimension one foliations is
parametrized by RP
7
∪{1 ,...,8}. This follows from Theorem 13.4.5 since all these
symmetric spaces have rank r = 8 an d trivial symmetry group A.
Another interesting special case is r = 1, that is, M is a hyperbolic space over one
of the normed real division algebras R, C, H or O. From Theorem 13.4.5 we see that
there are exactly two congruence classes of homogeneous codimension one foliations
of M.Thefirst one, coming from the 0-dimensional r eal projective space, is the well
known horosphere foliation. The second foliation is not so well known except for
the real hyperbolic case. In the case of RH
n
, we get the foliation whose leaves are
a totally geodesic RH
n−1
⊂ RH
n
and its equidistant hypersurfaces. Comparing this
with Cartan’s classification of homogeneous hypersurfaces in Theorem 13.5.2 we
see that we indeed got all homogeneous hypersurfaces of RH
n
that are not tubes
around a lower dimensional submanifold. In the case of CH
n
, the minimal orbit of
the second foliation is precisely the minimal homogeneous ruled real hypersurface of
CH
n
discovered b y Lohnherr [199]. The geometry of the second foliation has been
investigated for all hyperbolic spaces in [23]. We summarize this in the following.
Corollary 13.4.6 Up to isometric congruence, there are exactly two homogeneous
codimension one foliations of FH
n
= G/K, F ∈{R,C,H,O},wheren≥2 (and n = 2
if F = O):
(1) The horosphere foliation of FH
n
. This is the foliation of FH
n
whose leaves are
the orbits of the nilpotent group N is an Iwasawa decomposition of G = KAN.
(2) The foliation F
n
F
of FH
n
. This is the foliation of FH
n
whose leaves are the
orbits of the solvable group S
(see page 355). This foliation has exactly one
minimal leaf.
362 Submanifolds and Holonomy
13.5 Polar actions on hyperbolic spaces
13.5.1 Polar actions on real hyperbolic spaces
Polar actions on the real hyperbolic space RH
n
= SO
o
1,n
/SO
n
were first investi-
gated and classified by Wu [348]. We give here a description using parabolic subal-
gebras. Consider an Iwasawa descomposition
so
1,n
= k ⊕a ⊕n
of so
1,n
,where
k
∼
=
so
n
, a
∼
=
R , n = g
α
∼
=
R
n−1
.
The solvable subgroup AN of SO
o
1,n
with Lie algebr a a ⊕n acts simply transitively
on RH
n
. The orbits of the abelian subgroup N form a horosphere foliation of RH
n
and each orbit of N is isometric to a Euclidean space E
n−1
.
Let v be a k-dimensional (0 ≤ k ≤ n −2) linear subspace of n and denote by
v
⊥
= n v the orthogonal complement of v in n. The subspace a ⊕v is a subalgebra
of a ⊕n. The closed subgroup of AN with Lie algebra a ⊕v induces a Riemannian
foliation of RH
n
and the orbit containing o is a totally geodesic RH
k+1
⊂ RH
n
.The
orbits of the closed subgroup of N with Lie algebra v form a Riemannian foliation
of each horosphere (i.e., N-orbit) and the orbit containing o is a horosphere in the
totally geodesic RH
k+1
⊂ RH
n
.
Now let k
0
∼
=
so
n−1
be the centralizer of a in k.Thenq = k
0
⊕a ⊕n is a parabolic
subalgebra of so
1,n
.Leth
0
be a subalgebra of k
0
such that [h
0
,v] ⊂v and the subgroup
H
0
of K
0
∼
=
SO
n−1
is closed and acts polarly on v
⊥
∼
=
R
n−k−1
. Then both h
1
= h
0
⊕v
and h
2
= h
0
⊕a ⊕n are subalgebras of q and the corresponding closed subgroups H
1
and H
2
of the parabolic subgroup Q ⊂ SO
o
1,n
act polarly on RH
n
. By construction,
H
2
has a totally geodesic RH
k+1
⊂ RH
n
as an orbit and H
1
has a horosphere in this
totally geodesic RH
k+1
⊂ RH
n
as an orbit. We can also write H
1
= H
0
V and H
2
=
H
0
AV for these two groups, where V is the closed subgroup of n with Lie algeb ra v.
These are polar actions on RH
n
with a fixed point at infinity, as the corresponding
groups are contained in a parabolic subgroup of SO
o
1,n
.
There are also polar actions with a fixed point in RH
n
.LetH
3
be a closed sub-
group of K = SO
n
acting polarly on p
∼
=
T
o
RH
n
via the isotropy representation. Then
the action of H
3
on RH
n
is polar and has a fixed point at o.
Wu proved in [348] that every polar action on RH
n
is orbit equivalent to one of
the actions we just constructed.
Theorem 13.5.1 (Wu) Every polar action on RH
n
= SO
o
1,n
/SO
n
is orbit equivalent
to the action of the closed subgroup H of SO
o
1,n
with Lie algebra h such that
1. H has a fixed point at o (i.e., H is contained in the isotropy group K = SO
n
)
and H acts polarly on T
o
RH
n
∼
=
R
n
via the isotropy representation, or
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