348 Submanifolds and Holonomy
The Iwasawa decomposition of SL
r+1
(R) therefore describes the unique decompo-
sition of a matrix in SL
r+1
(R) into the product of an orthogonal matrix with deter-
minant one, a diagonal matrix with determinant one, and an upper triangular matrix
with entries equal to one in the diagonal. The solvable Lie group AN of upper trian-
gular matrices with deter minant one acts simply tran sitively on the symmetric space
M = SL
r+1
(R)/SO
r+1
. Thus, we have a natural identification of the Riemannian
symmetric space M = SL
r+1
(R)/SO
r+1
with the solvable Lie group
AN =
⎧
⎪
⎪
⎪
⎪
⎪
⎪
⎨
⎪
⎪
⎪
⎪
⎪
⎪
⎩
⎛
⎜
⎜
⎜
⎜
⎜
⎜
⎝
x
11
x
12
x
13
··· x
1,r+1
0 x
22
x
23
··· x
2,r+1
.
.
.
.
.
.
.
.
.
.
.
.
.
.
.
000
.
.
.
x
r,r+1
000··· x
r+1,r+1
⎞
⎟
⎟
⎟
⎟
⎟
⎟
⎠
: x
ij
∈R , x
11
·...·x
r+1,r+1
= 1
⎫
⎪
⎪
⎪
⎪
⎪
⎪
⎬
⎪
⎪
⎪
⎪
⎪
⎪
⎭
.
equipped with a suitable left- invariant Riemann ian metric.
The Dynkin diagram for M = SL
r+1
(R)/SO
r+1
is
α
1
α
2
α
r−1
α
r
and all multiplicities are equal to one.
We now discuss the parabolic subalgebras q
Φ
of sl
r+1
(R).
If Φ = /0, we get the minimal parabolic subalgebra q
/0
= a ⊕n, which is the
solvable subalgebra of sl
r+1
(R) consisting of all upper triangular (r + 1) ×(r + 1)-
matrices with trace zero.
If Φ = Λ,thenq
Λ
= sl
r+1
(R).
Now assume that Φ /∈{/0,Λ} and decompose Φ into Φ = Φ
1
∪...∪Φ
s
,where
each Φ
j
corresponds to a connected component of the Dynkin subdiagram induced
by Φ.Then
l
Φ
∼
=
sl
|Φ
1
|+1
(R) ⊕...⊕sl
|Φ
s
|+1
(R) ⊕R
r−|Φ|
with
m
Φ
= g
Φ
∼
=
sl
|Φ
1
|+1
(R) ⊕...⊕sl
|Φ
s
|+1
(R) and a
Φ
∼
=
R
r−|Φ|
.
This direct sum corresponds to the real vector space of (r + 1) ×(r + 1)-matrices
with nonzero entries only in a certain block diagonal decomposition. The nilpotent
subalgebra n
Φ
consists of strictly upper triangular matrices all of whose entries in
the intersection of the upper triangle and the blocks are zero. The parabolic subal-
gebra q
Φ
is then given by matrices with zero entries below a certain block diagonal
decomposition. So we can conclude that q
Φ
consists of all block diagonal upper tri-
angular matrices, where the block decomposition of the matrix corresponds to the
decomposition of Φ into connected subsets Φ
j
.
We finally come to the horospherical decompositions of M = SL
r+1
(R)/SO
r+1
.
If Φ = /0, then F
Φ
= R
r
, B
Φ
= {o},andN
Φ
is a maximal horocyclic subgroup of
SL
r+1
(R). The resulting horospherical decomposition is M = R
r
×N
Φ
.
If Φ = Λ,thenF
Φ
= M and we get a trivial horospherical decomposition.
Polar Actions on Symmetric Spaces of Noncompact Type 349
Now assume that Φ /∈{/0, Λ} and decompose Φ into Φ = Φ
1
∪...∪Φ
s
as above.
Then
B
Φ
∼
=
SL
|Φ
1
|+1
(R)/SO
|Φ
1
|+1
×...×SL
|Φ
s
|+1
(R)/SO
|Φ
s
|+1
and A
Φ
∼
=
R
r−|Φ|
.
The nilpotent group N
Φ
consists of upper triangular matrices as in n
Φ
but with entries
equal to one in the diagonal.
Example 13.2.2 (The symmetric space G
2
2
/SO
4
) The Riemannian symmetric space
M = G
2
2
/SO
4
has rank 2 and dimension 8. Its restricted root system is of type (G
2
) and the associ-
ated Dynkin diagram is
α
1
α
2
_jt
with all multiplicities equal to 1. The positive roots with respect to Λ = {
α
1
,
α
2
} are
Ψ
+
= {
α
1
,
α
2
,
α
1
+
α
2
,2
α
1
+
α
2
,3
α
1
+
α
2
,3
α
1
+ 2
α
2
}.
Since g
2
2
is a split real form of g
2
(C), the centralizer k
0
of a in k is trivial, an d hence
we have g
0
= a.
There are two nontrivial parabolic subalgebras of g
2
2
, which are associated with
Φ
1
= Λ {
α
1
} = {
α
2
} and Φ
2
= Λ {
α
2
} = {
α
1
}. We replace the index Φ
i
by i,so
B
i
= B
Φ
i
, and so on.
We first discuss the case of Φ
1
.Thenwehave
l
1
= g
−
α
2
⊕g
0
⊕g
α
2
∼
=
sl
2
(R) ⊕R
∼
=
m
1
⊕a
1
,
n
1
= g
α
1
⊕g
α
1
+
α
2
⊕g
2
α
1
+
α
2
⊕g
3
α
1
+
α
2
⊕g
3
α
1
+2
α
2
∼
=
R
5
.
The nilpotent Lie algebra n
1
is 3-step nilpotent and generated by g
α
1
⊕g
α
1
+
α
2
.The
reductive Lie algebr a m
1
is semisimple and hence m
1
= g
1
. The parabolic subalgebra
q
1
= l
1
⊕n
1
is 9-dimensional.
The boundary component B
1
= M
1
·o = G
1
·o is isometric to a real hyperbolic
plane RH
2
= SL
2
(R)/SO
2
and the totally geodesic submanifold F
1
is isometric to
the Riemannian product B
1
×R = RH
2
×R. The horosph e rical decomposition o f
G
2
2
/SO
4
induced by the parabolic subalgebra q
1
therefore is
G
2
2
/SO
4
= RH
2
×R ×N
1
,
where N
1
is the 5-dimensional 3-step nilpotent subgroup of G
2
2
with Lie algebra n
1
.
Next, we discuss the case of Φ
2
.Inthiscasewehave
l
2
= g
−
α
1
⊕g
0
⊕g
α
1
∼
=
sl
2
(R) ⊕R
∼
=
m
2
⊕a
2
,
n
2
= g
α
2
⊕g
α
1
+
α
2
⊕g
2
α
1
+
α
2
⊕g
3
α
1
+
α
2
⊕g
3
α
1
+2
α
2
∼
=
R
5
.
350 Submanifolds and Holonomy
The nilpotent Lie algebra n
2
is 2-step nilpotent and generated by g
α
2
⊕g
α
1
+
α
2
⊕
g
2
α
1
+
α
2
⊕g
3
α
1
+
α
2
. In fact, n
2
is isomorphic to the 5-dimensional Heisenberg algebra
with one-dimensional center. Again, the reductive Lie algebr a m
2
is semisimple and
hence m
2
= g
2
. The parabolic subalgebra q
2
= l
2
⊕n
2
is also 9-dimensional, but we
can easily see that q
1
and q
2
are not isomorphic since their derived algebras n
1
and
n
2
are non-isomorphic nilpotent Lie algebras.
The boundary component B
2
= M
2
·o = G
2
·o is isometric to a real hyperbolic
plane RH
2
= SL
2
(R)/SO
2
and the totally geodesic submanifold F
2
is isometric to
the Riemannian product B
2
×R = RH
2
×R. The two boundary components B
1
and
B
2
appear to be similar, but they are not congruent or even isometric to each other.
One can show that the two real hyperbolic planes B
1
and B
2
have different Gaus-
sian curvature. This is a consequence of the algebraic property that the two simple
roots
α
1
and
α
2
have different length s. The horospherical decomposition of G
2
2
/SO
4
induced by the parabolic subalgebra q
2
is
G
2
2
/SO
4
= RH
2
×R ×N
2
,
where N
2
is the 2-step nilpotent subgroup of G
2
2
with Lie algebra n
2
. The nilpotent
Lie group N
2
is isomorphic to the 5-dimensional Heisenberg group.
Example 13.2.3 (The symmetric space G
2
(C)/G
2
) The symmetric space
M = G
2
(C)/G
2
has rank 2 and dimension 14. Its restricted root system is of type (G
2
) and the asso-
ciated Dynkin diagram is
α
1
α
2
_jt
with all multiplicities equal to 2. So, in terms of restricted roots, the only difference
between the symmetric spaces G
2
2
/SO
4
and G
2
(C)/G
2
are the mu ltiplicities of the
roots, which are equal to 1 for G
2
2
/SO
4
and equal to 2 for G
2
(C)/G
2
.Now,inthe
case of G
2
(C)/G
2
, the centralizer k
0
of a in k is isomorphic to u
1
⊕u
1
.Usingthe
analogous notation as in the previous example, we get for the case of Φ
1
:
l
1
= g
−
α
2
⊕g
0
⊕g
α
2
∼
=
gl
2
(C)
∼
=
(sl
2
(C) ⊕u
1
) ⊕R
∼
=
m
1
⊕a
1
,
n
1
= g
α
1
⊕g
α
1
+
α
2
⊕g
2
α
1
+
α
2
⊕g
3
α
1
+
α
2
⊕g
3
α
1
+2
α
2
∼
=
C
5
.
The nilpotent Lie algebra n
1
is 3-step nilpotent and generated by g
α
1
⊕g
α
1
+
α
2
.In
contrast to the previous case of G
2
2
/SO
4
, the reductive Lie algebra m
1
is not semisim-
ple and splits as m
1
= g
1
⊕z
1
= sl
2
(C) ⊕u
1
. The parabolic subalgebra q
1
= l
1
⊕n
1
of g
2
(C) is 18-dimensional.
The boundary component B
1
= M
1
·o = G
1
·o is isometric to a real hyperbolic
space RH
3
= SL
2
(C)/SU
2
and the totally geodesic submanifold F
1
is isometric to the
Riemannian product B
1
×R = RH
3
×R. The unitary group U
1
in M
1
corresponding
to the center z
1
= u
1
of m
1
acts trivially on B
1
= RH
3
. The horospherical decompo-
sition of G
2
(C)/G
2
induced by the parabolic subalgebra q
1
therefore is
G
2
(C)/G
2
= RH
3
×R ×N
1
,
Polar Actions on Symmetric Spaces of Noncompact Type 351
where N
1
is the 10-dimensional 3-step nilpotent subgroup of G
2
(C) with Lie algebra
n
1
.
In case of Φ
2
we have a similar outcome, where the nilpotent Lie algebra n
2
is
2-step nilpotent. The boundary component B
2
= M
2
·o is isometric to a real hyper-
bolic space RH
3
= SL
2
(C)/SU
2
which has different sectional curvature than B
1
.The
horospherical decomposition of G
2
(C)/G
2
induced by the p arabolic subalgebra q
2
is
G
2
(C)/G
2
= RH
3
×R ×N
2
,
where N
2
is the 10-dimensional 2-step nilpotent subgroup of G
2
(C) with Lie algebra
n
2
.
13.3 Polar actions without singular orbits
We saw in Theorem 12.2.1 that 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. This is not true in the noncompact situation. The simplest
examples of polar actions which are not hyperpolar can be constructed from the
Chevalley decomposition of parabolic subgroups. These polar actions do not have
any singular orbits, and this why we first focus on the case of polar actions without
singular orbits. The orbits of a polar action without singular orbits form a homoge-
neous Riemannian foliation. We say that a homogeneous Riemannian foliation of a
Riemannian manifold is polar (respectively hyperpolar) if the leaves of the foliation
are the orbits of a polar (respectively hyperpolar) action on the manifold.
We start with an algebraic characterization of polar actions without singular or-
bits on Riemannian symmetric spaces of noncompact type (see [137] for the analo-
gous statement in the compact case).
Proposition 13.3.1 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 polar if and only if
(i) h
⊥
p
is a Lie triple system in p, and
(ii) h is orthogonal to the subalgebra [h
⊥
p
,h
⊥
p
] ⊕h
⊥
p
of g.
Assume that the action of H on M is polar and let H
⊥
p
be the connected subgroup
of G with Lie algebra [h
⊥
p
,h
⊥
p
] ⊕h
⊥
p
. Then the orbit Σ = H
⊥
p
·oisasectionofthe
H-actiononM.
352 Submanifolds and Holonomy
Proof If the action of H on M is polar, then h
⊥
p
is a Lie triple system by definition
of polar action. Assume that h
⊥
p
is a Lie triple system, which implies that Σ is a
connected complete totally geodesic submanifold of M. We have to show that the
action of H on M is polar if and only if h,[h
⊥
p
,h
⊥
p
]⊕h
⊥
p
= {0}.Letp ∈ M such that
p /∈H ·o.SinceH ·o is a c losed submanifold of M, there exists a point q ∈H ·o such
that the distance b between p and q is equal to the distance between p and H ·o.Let
γ
be a unit speed geodesic in M with
γ
(0)=q and
γ
(b)=p. A standard variational
argument shows that
˙
γ
(0),T
q
(H ·o)= {0}.ThisimpliesthatΣ intersects each orbit
of the H-action on M.SinceH induces a foliation, it therefore remains to show that
T
p
(H · p),T
p
Σ = {0} holds for all p ∈ Σ if and only if h,[h
⊥
p
,h
⊥
p
] ⊕h
⊥
p
= {0}.
Let
γ
be a geodesic in Σ with
γ
(0)=o and 0 =
˙
γ
(0)=
ξ
∈ h
⊥
p
.ForX ∈ h and
η
∈h
⊥
p
we denote by X
∗
and
η
∗
the Killing vector fields on M that are induced from
X and
η
, respectively. Then we have
T
γ
(t )
(H ·
γ
(t)) = {X
∗
γ
(t )
: X ∈ h} and T
γ
(t )
Σ = {
η
∗
γ
(t )
:
η
∈ h
⊥
p
}.
The restrictions X
∗
γ
and
η
∗
γ
of two such Killing vector fields X
∗
and
η
∗
to
γ
satisfy
d
dt
t=0
X
∗
γ
(t )
,
η
∗
γ
(t )
= [
ξ
∗
,X
∗
]
o
,
η
∗
o
+ X
∗
o
,[
ξ
∗
,
η
∗
]
o
= −[
ξ
,
η
],X,
using the facts that [
ξ
∗
,X
∗
]=−[
ξ
,X]
∗
, [
ξ
∗
,
η
∗
]=−[
ξ
,
η
]
∗
, [
ξ
,
η
] ∈k, and that ad(
ξ
)
is a selfadjoint endomorphism on g. It follows that h,[h
⊥
p
,h
⊥
p
]⊕h
⊥
p
= {0}if T
p
(H ·
p),T
p
Σ= {0} holds for all p ∈ Σ.
Conversely, assume that h, [h
⊥
p
,h
⊥
p
] ⊕h
⊥
p
= {0}. Then, for each X ∈ h,there-
striction X
∗
γ
of the Killing vector field X
∗
to
γ
is the Jacobi field along
γ
with initial
values
X
∗
γ
(0)=X
∗
o
= X
p
∈h
p
and (X
∗
γ
)
(0)=[
ξ
∗
,X
∗
]
o
= −[
ξ
,X]
∗
o
= −[
ξ
,X]
p
∈h
p
,
where the subscript (·)
p
indicates orthogonal projection onto p. Since both initial
values of the Jacobi field X
∗
γ
are in h
p
=
ν
o
Σ, it follows that X
∗
γ
takes values in the
normal bundle of Σ along
γ
. This implies T
γ
(t )
(H ·
γ
(t)),T
γ
(t )
Σ = {0} for all t ∈ R.
Since this holds for every geodesic
γ
in Σ with
γ
(0)=o and 0 =
˙
γ
(0)=
ξ
∈ h
⊥
p
,we
conclude that T
p
(H · p),T
p
Σ = {0} holds for all p ∈ Σ.
We will now use Proposition 13.3.1 for the construction of polar actions on Rie-
mannian symmetric spaces of noncompact type which are not hyperpolar. For this
we are going to use th e general theory of parabolic subalgebr as that we outlined in
Section 13.2. Let M = G/K be a Riemannian symmetric space of noncompact type
and let g = k ⊕p be the corresponding Cartan decomposition. Let q
Φ
be a parabolic
subalgebra of g and c onsider its Chevalley decomposition
q
Φ
= l
Φ
⊕n
Φ
.
The orbits of the action of the nilpotent subgroup N
Φ
of G form a Riemannian folia-
tion of M without singular orbits. First, we already know from (13.1) that
(n
Φ
)
⊥
p
= l
Φ
∩p = p
Φ
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