Polar Actions on Symmetric Spaces of Noncompact Type 353
is a Lie triple system and from (13.2) that the corresponding connected complete
totally geodesic submanifold is
Σ = L
Φ
·o = F
Φ
= B
Φ
×E
r−|Φ|
.
We a lso have
[(n
Φ
)
⊥
p
,(n
Φ
)
⊥
p
]=[p
Φ
,p
Φ
] ⊂ k
Φ
= l
Φ
∩k ⊂ l
Φ
,
which implies
[(n
Φ
)
⊥
p
,(n
Φ
)
⊥
p
] ⊕(n
Φ
)
⊥
p
⊂ l
Φ
.
Since n
Φ
,l
Φ
= {0}, we can apply Proposition 13.3.1 and conclude that the action
of N
Φ
on M is polar and Σ = L
Φ
·o = F
Φ
= B
Φ
×E
r−|Φ|
is a section of this action.
Obviously, Σ is flat if and only if Φ = /0. Thus we have proved:
Proposition 13.3.2 Let M = G/K be a Riemannian symmetric space of noncompact
type and Q
Φ
= L
Φ
N
Φ
be the Chevalley d ecomposition of a parabolic subgroup Q
Φ
of G. Then the action of N
Φ
on M is polar. Moreover, the action of N
Φ
is hyperpolar
if and only if Q
Φ
is a minimal parabolic subgroup of G.
Thus any irreducible Riemannian symmetric space of noncompact type of rank
≥ 2 admits a polar and non-hyperpolar action with an orbit of positive dimension.
As a consequence we see that the assumption of compactness in Theorem 12.2.1 is
essential. The above proposition can be generalized as follows (see [30]). Let Q
Φ
=
M
Φ
A
Φ
N
Φ
be the Langlands decomposition of a parabolic subgroup Q
Φ
of G.The
abelian subgroup A
Φ
is isomorphic to the abelian Lie group R
r−|Φ|
andsothereis
a natural one-to-one correspondence between linear subspaces of R
r−|Φ|
and linear
subspaces of A
Φ
. In particular, the orthogonal complement V
⊥
of a linear subspace V
in A
Φ
can be identified with the orthogonal complement of the corresponding linear
subspace in R
r−|Φ|
.
Theorem 13.3.3 Let M = G/K be a Riemannian symmetric space of noncompact
type and Q
Φ
= M
Φ
A
Φ
N
Φ
be the Langlands decomposition of a parabolic subgroup
Q
Φ
of G. Let V be a linear subspace of A
Φ
and assume that (Φ,V ) =(/0,A
Φ
).Then
the action of V N
Φ
⊂ A
Φ
N
Φ
on M is polar and Σ = B
Φ
×V
⊥
=(M
Φ
·o) ×(V
⊥
·o) is
a section of this action. Moreover, the action of V N
Φ
on M is hyperpolar if and only
if Q
Φ
is a minimal parabolic subgroup of G.
Proof Exercise 13.8.1.
The choice of V = {0} in Theorem 13.3.3 leads to Proposition 13.3.2.
13.4 Hyperpolar actions without singular orbits
In this section we discuss hyperpolar actions without singular orbits on Riemann-
ian symmetric spaces of noncompact type. The algebraic characterization of polar
354 Submanifolds and Holonomy
actions without singular orbits given in Proposition 13.3.1 simplifies considerably in
this context.
Corollary 13.4.1 Let M = G/K be a Riemannian symmetric space of noncompact
type and H be a connected closed subgroup of G whose orbits form a Riemannian
foliation of M (without singular orbits). Consider the corresponding Cartan decom-
position g = k ⊕p and define
h
⊥
p
= {
ξ
∈p :
ξ
,Y = 0 for all Y ∈h}.
Then the action of H on M is hyperpolar if and only if h
⊥
p
is an abelian subspace
of p. Assume that the action of H on M is hyperpolar and let H
⊥
p
be the connected
subgroup of G with Lie algebra h
⊥
p
. Then the orbit Σ = H
⊥
p
·o is a section of the
H-actiononM.
We will now outline the classification by Berndt, D´ıaz-Ramos and Tamaru in [30]
of hyperpolar actions without singular orbits on Riemannian symmetric spaces of
noncompact type, up to orbit equivalence. The classification, up to orbit equivalence,
of hyperpolar actions without singular orbits is equivalent to the classification of
hyperpolar homogeneous foliations, up to isometric congruence. It turns out that hy-
perpolar homogeneous foliations on Riemannian symmetric spaces of noncompact
type are combinations of certain model foliations which are combined using horo-
spherical decompositions. We will first describe these model foliations.
The totally geodesic subspaces of the Euclidean space E
m
are the affine subspaces
of the underlying vector space. Since affine subspaces are flat, “polar” and “hyper-
polar” have the same meaning in the Euclidean setting. For each linear subspace V
of E
m
we define a foliation F
m
V
on E
m
by
(F
m
V
)
p
= p +V = {p + v : v ∈V }
for all p ∈ E
m
. Geometrically, the leaves of the foliation F
m
V
are the affine subspaces
of E
m
which are parallel to V . It is obvious that F
m
V
is a hyperpolar homogeneous
foliation of E
m
. Every hyperpolar homogeneous foliation of E
m
is isometrically con-
gruent to F
m
V
for some linear subspace V .
Let M = G/K be a hyperbolic space over a normed real division algebra F ∈
{R,C,H,O}. We denote such a hyperbolic space by FH
n
,wheren ≥ 2 is the dim-
ension of the manifold over the algebra F,andn = 2ifF = O.AsM has rank one,
there is exactly one simple root
α
and
g = g
−2
α
⊕g
−
α
⊕g
0
⊕g
α
⊕g
2
α
is the restr icted root space decomposition of g .IfF = R,theng
±2
α
= {0 }.The
subalgebra n = g
α
⊕g
2
α
of g is nilpotent and a ⊕n is a solvable subalgebra of g.The
vector space decomposition g = k ⊕a ⊕n is an Iwasawa decomposition of g.Since
M is isometric to the solvable Lie group AN equipped with a suitable left- invariant
Riemannian metric, it is obvious that every subalgebra of a ⊕n of codimension one
Polar Actions on Symmetric Spaces of Noncompact Type 355
induces a homogeneous codimension one foliation of M.Let be a one-dimensional
linear subspace of g
α
.Thens
= a ⊕(n ) is a subalgebra of a ⊕n of codimension
one. The orbits of the corresponding connected closed subgroup S
of AN form a
homogeneous foliation
F
n
F
= union of the orbits of the action of S
on FH
n
.
Different choices of lead to isometrically congruent foliations, because the central-
izer K
0
of a in K acts transitively on the unit sphere in g
α
. The leaves of F
n
R
are
a totally geodesic hyperplane RH
n−1
in RH
n
and the equidistant hypersurfaces. If
F = C, one of the leaves of F
n
C
is the minimal ruled real hypersurface in CH
n
gen-
erated by a horocycle in a totally geodesic RH
2
⊂ CH
n
and the other leaves are the
equidistant hypersurfaces.
Let
M = F
1
H
n
1
×...×F
k
H
n
k
be the Riemannian product of k hyperbolic spaces, where k ≥ 2 is a positive integer
and F
k
∈{R,C,H,O}.Then
F
n
1
F
1
×...×F
n
k
F
k
is a hyperpolar homogeneous foliation of M. This is an elementary consequence of
the previous example.
Let
M = F
1
H
n
1
×...×F
k
H
n
k
×E
m
be the Riemannian product of k hyperbolic spaces and an m-dimensional Euclidean
space, where k and m are positive integers. Moreover, let V be a linear subspace of
E
m
.Then
F
n
1
F
1
×...×F
n
k
F
k
×F
m
V
is a hyperpolar homogeneous foliation of M.
We now come to the general situation. Let M be a Riemannian symmetric space
of noncompact type and Φ be a subset of Λ with the property that any two roots in
Φ are not connected in the Dynkin diagram of the restricted root system associated
with M. In other words, any two distinct roots in Φ are orthogonal. We call such a
subset Φ an orthogonal subset of Λ. Each simple root
α
∈ Φ determines a totally
geodesic hyperbolic space F
α
H
n
α
⊂ M. In fact, F
α
H
n
α
⊂M is the orbit of G
Φ
, Φ =
{
α
}, containing the point o.If2
α
/∈ Ψ, that is, if the vertex in the Dynkin diagram
corresponding to
α
is of the form
 
,thenF
α
= R and the dimension n
α
is equal to
m
α
+ 1, where m
α
is the multiplicity of the root
α
.If2
α
∈Ψ, that is, if the vertex in
the Dynkin diagram corresponding to
α
is of the form
   
,thenF
α
∈{C,H,O}.Note
that this can happen only if Ψ is of type (BC
r
) and
α
=
α
r
.Wehavem
2
α
∈{1 , 3, 7},
F
α
=
⎧
⎪
⎨
⎪
⎩
C if m
2
α
= 1,
H if m
2
α
= 3,
O if m
2
α
= 7,
356 Submanifolds and Holonomy
and
n
α
=
⎧
⎪
⎨
⎪
⎩
m
α
2
+ 1ifm
2
α
= 1,
m
α
4
+ 1ifm
2
α
= 3,
2ifm
2
α
= 7.
The symmetric sp ace F
Φ
in the horo sp herical decomposition of M induced from Φ
is isometric to the Riemannian product of |Φ| hyperbolic spaces and an (r −|Φ|)-
dimensional Euclidean space, that is,
F
Φ
= B
Φ
×E
r−|Φ|
∼
=
∏
α
∈Φ
F
α
H
n
α
×E
r−|Φ|
.
Then
F
Φ
=
∏
α
∈Φ
F
n
α
F
α
.
is a hyperpolar homogeneous foliation of B
Φ
.LetV be a linear subspace of E
r−|Φ|
.
Then
F
Φ,V
= F
Φ
×F
r−|Φ|
V
×N
Φ
⊂ B
Φ
×E
r−|Φ|
×N
Φ
= F
Φ
×N
Φ
∼
=
M
is a homogeneous foliation of M.
Each f oliation F
n
α
F
α
of F
α
H
n
α
corresponds to a subalgebra of g
{
α
}
of the form
a
{
α
}
⊕(g
α
α
) ⊕g
2
α
with some one-dimensional linear subspace
α
of g
α
. Thus
the foliation F
Φ
of B
Φ
corresponds to the subalgebra
a
Φ
⊕(n
Φ
Φ
) ⊂ g
Φ
⊂ m
Φ
,
where
n
Φ
=
α
∈Φ
(g
α
⊕g
2
α
) and
Φ
=
α
∈Φ
α
.
Note that a
Φ
⊕n
Φ
is the solvable subalgebra in an Iwasawa deco mposition of the
semisimple Lie algebra g
Φ
. Therefore, the foliation F
Φ,V
of M corresponds to the
subalgebra
s
Φ,V
=(a
Φ
⊕(n
Φ
Φ
)) ⊕V ⊕n
Φ
⊂ m
Φ
⊕a
Φ
⊕n
Φ
= q
Φ
of q
Φ
, where we identify canonically V ⊂ E
r−|Φ|
= A
Φ
·o with the corresponding
subspace of a
Φ
. One can show that different choices of
α
in g
α
lead to isometrically
congruent foliations F
Φ,V
of M.
We are now in a position to formulate the main classification result.
Theorem 13.4.2 (Berndt, D´ıaz-Ramos, Tamaru) Let M be a connected Riemann-
ian symmetric space of noncompact type.
(i) Let Φ be an orthogonal subset of Λ and V be a linear subspace of E
r−|Φ|
.Then
F
Φ,V
= F
Φ
×F
r−|Φ|
V
×N
Φ
⊂ B
Φ
×E
r−|Φ|
×N
Φ
= M
is a hyperpolar homogeneous foliation of M.
Polar Actions on Symmetric Spaces of Noncompact Type 357
(ii) Every hyperpolar homogeneous foliation of M is isometrically congruent to
F
Φ,V
for some orthogonal subset Φ of Λ and some linear subspace V of E
r−|Φ|
.
For the proof we refer to [30]. Theorem 13.4.2 tells us how to construct, up
to orbit equivalence, all hyperpolar actions without singular orbits on Riemann-
ian symmetric spaces of noncompact type. We will discuss this now for M =
SL
r+1
(R)/SO
r+1
.
Example 13.4.1 (Homogeneous hyperpolar foliations of SL
r+1
(R)/SO
r+1
) The
Dynkin d iagram associated with M = SL
r+1
(R)/SO
r+1
is
       
α
1
α
2
α
r−1
α
r
and therefore the orthogonal subsets Φ of Λ correspond precisely to the subsets of
{1,...,r} containing no two adjacent positive integers. Let k = |Φ| be the cardinality
of an orthogonal subset Φ of Λ. Since the multiplicity of each simple root is one,
the semisimple part B
Φ
of the ho rospherical decomposition M
∼
=
B
Φ
×E
r−k
×N
Φ
is isometric to the Riemannian product of k real hyperbolic planes RH
2
. On each
of these real hyperbolic planes we choose the foliation determined by a geodesic
and its equidistant curves. The product of these foliations determines the foliation
F
Φ
of the k-fold product B
Φ
of real hyperbolic planes. On the abelian part E
r−k
we
choose a foliation F
r−k
V
by parallel affine subspaces (including the trivial foliations
of dimension 0 and r −k). The product foliation F
Φ
×F
r−k
V
of the totally geodesic
submanifold F
Φ
= B
Φ
×E
r−k
of M is hyperpolar. The foliation F
Φ,V
is then obtained
by taking the product of this foliation with the horocycle foliation N
Φ
of M. Theorem
13.4.2 says that every hyperpolar foliation of M = SL
r+1
(R)/SO
r+1
is obtained in
this way.
The corank of the foliation F
Φ,V
equals the cohomogeneity of the corresponding
hyperpolar action by the subgroup S
Φ,V
with Lie algebra s
Φ,V
. Since every coho-
mogeneity one action is hyperpolar, Theorem 13.4.2 includes the classification of
cohomogeneity one actions without singular orbits, or equivalently, of homogeneous
codimension one foliations. The homogeneous codimension one foliations of irre-
ducible Riemannian symmetric spaces of noncompact type were classified earlier by
Berndt and Tamaru in [37] with different m ethods. We will discuss this special case
now in more detail.
We denote b y A the group of symmetries of the Dynkin diagram associated with
the root system (R) of M. There are just three possibilities, namely
A =
⎧
⎪
⎨
⎪
⎩
A
3
,ifR = D
4
A
2
,ifR ∈{A
r
(r ≥ 2), D
r
(r ≥ 5), E
6
},
A
1
,otherwise.
where A
k
is the symmetric group on a set of k elements. The first two cases corre-
spond to triality and duality principles on the symmetric space that were discovered
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