Submanifold Geometry of Orbits 57
Proof Without loss of generality we can assume that the homogeneous submani-
fold, say G · p, is irreducible. By Theorem 2.5.1 and its proof there exists a basis of
L (
ρ
(G)) of the form (A
1
,d
1
),...,(A
n
,d
n
), where all vectors d
i
belong to the inter-
section V of the kernels of A
1
,...,A
n
. We assume that the first l < n vectors are in
the isotropy algebra at p and that A
l+1
p + d
l+1
,...,A
n
p + d
n
form an orthonormal
basis of T
p
(G · p). Now decompose p into p = p
1
+ p
2
with p
1
∈V
⊥
and p
2
∈V (so
A
i
p
2
= 0andd
i
= 0fori = 1,...,l). For i = l + 1,...,n we define
γ
i
(t)=Exp(t(A
i
,d
i
))p = e
tA
i
p
1
+ td
i
+ p
2
.
Note that p
1
belongs to the normal space of G ·p at p,sinced
i
∈V and A
i
is skewsym-
metric. We claim that p
1
= 0. Using minimality we get
0 =
n
∑
i=l+1
γ
i
(0), p
1
=
n
∑
i=l+1
A
2
i
p
1
, p
1
=
n
∑
i=1
A
2
i
p
1
, p
1
=
n
∑
i=1
A
i
p
1
,A
i
p
1
and hence A
i
p
1
= 0foralli = 1,...,n and p
1
∈V . This implies p
1
= 0sincep
1
also
belongs to V
⊥
. It is now clear that G · p = G ·(0, p
2
) coincides with the linear span
of d
l+1
,...,d
n
and is totally geodesic.
This result is sharp in the sense that there exist minimal submanifolds of R
n
with
codimension one (for instance, minimal surfaces of revolution). By Calabi’s Rigidity
Theorem, any holomorphic isometry of a complex submanifold of C
n
extends to C
n
.
On the other hand, any complex submanifold is minimal. So we have the following:
Corollary 2.5.3 (Di Scala) Every complex homogeneous submanifold of C
n
is to-
tally geodesic.
2.6 Homogeneous submanifolds of hyperbolic spaces
In this section we outline the results by Di Scala and Olmos in [107] about ho-
mogeneous submanifolds of the real hyperbolic space H
n
= SO
o
n,1
/SO
n
.Wefirst
introduce some notation. Let (V , ·, ·) be a real vector space endowed with a non-
degenerate symmetric bilinear form of signature (n,1). We can identify V with the
Lorentzian space R
n,1
and hence Aut(V, ·,·)
∼
=
O
n,1
. It is well-known that H
n
can be
identified with a connected component of the set of points p ∈R
n,1
with p, p= −1.
Note that the identity component SO
o
n,1
of O
n,1
acts transitively on H
n
by isome-
tries. An affine subspace W of V is called Riemannian, Lorentzian, or degenerate
if the restriction of ·,· to the vector part of W is positive definite, has signature
(dimW −1,1) or is degenerate, respectively. A horosphere in H
n
is a hypersurface
that is obtained by intersecting H
n
with an affine degenerate hyperplane. Recall that
the ideal boundary H
n
(∞) is the set of equivalence classes of asymptotic geodesics.
Thus H
n
(∞) can be regarded as the set of light lines through the origin, that is,
58 Submanifolds and Holonomy
H
n
(∞) ≡{z ∈ R
n,1
: z,z = 0}. Observe that a point z at infinity defines a folia-
tion of H
n
by equidistant horospheres. We say that the horosphere Q is center e d a t
z ∈ H
n
(∞) if Q is a leaf of that foliation. An action of a subgroup G of O
n,1
is called
weakly irreducible if it leaves invariant degenerate subspaces only.
The classification of homogeneous submanifolds of H
n
is basically given by the
following result.
Theorem 2.6.1 (Di Scala, Olmos) Let G be a connected (not necessarily closed) Lie
subgroup of I
o
(H
n
)=SO
o
n,1
and consider the action of G on H
n
. Then one of the
following statements holds:
(1) G has a fixed point;
(2) G has a unique nontrivial totally geodesic orbit (possibly H
n
);
(3) All orbits are contained in horospheres centered at the same point at infinity.
This result is also a tool for the proof of the next result, which shows how the
theory of homogeneous submanifolds of H
n
can be used to obtain general results
about the action of a connected Lie subgroup of O
n,1
on the Lorentzian space R
n,1
.
Theorem 2.6.2 (Di Scala, Olmos) Let G be a connected (not necessarily closed) Lie
subgroup of SO
o
n,1
and assume that the action of G on the Lorentzian space R
n,1
is
weakly irreducible. Then G acts transitively either on H
n
or on a horosphere of H
n
.
Moreover, if G acts irreducibly, then G = SO
o
n,1
.
Theorem 2.6.2 has an immediate consequence, which provides a purely geomet-
ric answer to a question posed in [16].
Corollary 2.6.3 (Berger [17, 18]) Let M be an n-dimensional Lorentzian manifold.
If the restricted holonomy group of M acts irreducibly, then it coincides with SO
o
n,1
.
In particular, if M is locally symmetric, then M has constant sectional curvature.
We will now present some ideas for the proof of Theorem 2.6.2. The fundamental
tools for the proof of Theorem 2.6.2 are Theorem 2.6.1 and the following uniqueness
result.
Lemma 2.6.4 Let G be a connected Lie subgroup of SO
o
n,1
. If the action of G on H
n
has a totally geodesic orbit, then no other orbit of positive dimension is minimal.
Proof Suppose that G · p is a totally geodesic orbit and consider another orbit G ·q
with G ·q = {q}.Let
γ
be a geodesic in H
n
minimizing the distance between q and
G · p. We may assume that
γ
meets G · p at p and that
γ
is parametrized so that
γ
(0)=p and
γ
(1)=q.Since
γ
minimizes the distance between q and G · p,
˙
γ
(0) is
perpendicular to G · p at p,thatis,
˙
γ
(0),X
∗
p
= 0forallX ∈ g,whereX
∗
denotes
as usual the Killing vector field generated by X .SinceX
∗
is a Killing vector field,
we have ∇
˙
γ
(t )
X
∗
,
˙
γ
(t) = 0forallt. This implies
d
dt
X
∗
γ
(t )
,
˙
γ
(t) = 0 and therefore
Submanifold Geometry of Orbits 59
X
∗
γ
(t )
,
˙
γ
(t) = 0forallt. This means that
˙
γ
(t) is orthogonal to the orbit G ·
γ
(t) at
γ
(t) for each t.
Now assume that X ∈ g satisfies X
∗
q
= 0andletΦ
X
∗
s
be the one-parameter group
of isometries generated by X
∗
.Leth : [0,1]×R → H
n
be defined by h
s
(t)=h(s,t)=
Φ
X
∗
s
(
γ
(t)). Note that X
∗
h
s
(t )
=
∂
∂
s
h
s
(t) and that h
s
(t) is a geodesic for each s.LetA
˙
γ
(t )
be the shape operator of the submanifold G ·
γ
(t) at
γ
(t) with respect to
˙
γ
(t) and
define
f (t)=−A
˙
γ
(t )
X
∗
γ
(t )
,
˙
γ
(t) =
D
∂
s
∂
∂
t
h
s
(t),X
∗
h
s
(t )
|
s=0
.
The derivative f
(t) of f (t) can be computed by
f
(t)=
D
∂
t
D
∂
s
∂
∂
t
h
s
(t),X
∗
h
s
(t )
|
s=0
+
D
∂
s
∂
∂
t
h
s
(t),
D
∂
t
X
∗
h
s
(t )
|
s=0
= R
∂
∂
t
h
s
(t),
∂
∂
s
h
s
(t)
∂
∂
t
h
s
(t),X
∗
h
s
(t )
|
s=0
+
D
∂
t
∂
∂
s
h
s
(t),
D
∂
t
X
∗
h
s
(t )
|
s=0
= R
˙
γ
(t),X
∗
γ
(t )
˙
γ
(t),X
∗
γ
(t )
+ ∇
˙
γ
(t )
X
∗
2
=
˙
γ
(t)
2
X
∗
γ
(t )
2
+ ∇
˙
γ
(t )
X
∗
2
.
This implies f
(t) ≥ 0and f
(1) > 0sinceX
∗
q
= 0. As G · p is totally geodesic, we
have f (0)=0 and hence A
˙
γ
(1)
X
∗
q
,X
∗
q
= −f (1) < 0. Thus A
˙
γ
(1)
is negative definite,
which shows that G ·q is not minimal.
As a consequence from Theorem 2.6.1 and Lemma 2.6.4 we see that every min-
imal homogeneous submanifold of H
n
is totally geodesic. As we saw in Theorem
2.5.2, the analogous statement for R
n
is also true.
A key fact in the proof of Theorem 2.6.1 is the following observation: If a normal
subgroup H of G has a totally geodesic orbit H · p with positive dimension, then
G · p = H · p. This is because G permutes the H-orbits, and hence H · p = G · p by
Lemma 2.6.4.
The next step for proving Theorem 2.6.1 is to study the two following cases
separately: G is semisimple (and of noncompact type) and G is not semisimple.
If G is not semisimple, one proves the statement first for abelian groups. Apply-
ing the previous observation to a normal abelian subgroup of G, three possibilities
can occur: G translates a geodesic, G fixes a point at infinity, or G has a proper
totally geodesic orbit. Consequently, a connected Lie subgroup G of O
n,1
that acts
irreducibly on R
n,1
must be semisimple. Finally, one shows that if G has a fixed point
z at infinity, then G has a totally geodesic orbit (possibly the entire H
n
), or G has a
fixed point in H
n
, or all of its orbits are contained in the horospheres centered at z.
TheideaisthatifG has neither a fixed point nor orbits in horospheres, then there
exists a codimension one subgroup H of G such that every H-orbit is contained in
the horosphere foliation determined by z.ThenH acts isometrically on horospheres,
and one can use the fact that H must have a totally geodesic orbit in each horosphere,
because each horosphere is a Euclidean space. At last, it is not hard to show that the
60 Submanifolds and Holonomy
union of all these totally geodesic orbits over all horospheres is a totally geodesic
G-invariant submanifold of H
n
.
When G is a semisimple Lie group, we choose an Iwasawa decomposition G =
KAN. Then one can show that the solvable subgroup AN of G has a minimal o rbit that
is also a G-orbit. One first chooses a fixed point p of the compact group K,which
always exists by Cartan’s Fixed Point Theorem. One can prove that the isotropy
group G
p
of G at p coincides with K. Then the mean curvature vector field of G ·p =
AN ·p is invariant under G
p
and if it does not vanish, then the G-orbits through points
on normal K-invariant geodesics are homothetic to the orbit G · p. These orbits are
also AN-orbits. The volume element of these orbits can be controlled by Jacobi vector
fields, eventually p roving that there exists a minimal G-orbit that is also an AN-orbit.
An induction argument involving n and the dimension of G completes the proof
of Theorem 2.6.1.
The idea for the proof of Theorem 2.6.2 is as follows: If G acts in a weakly
irreducible way, then the G-orbits must be contained in horospheres and, if an orbit is
a proper submanifold of some horosphere, one can construct a proper totally geodesic
G-invariant submanifold as the union of orbits parallel to totally geodesics orbits of
the action of G restricted to the horosphere. This is a contradiction because totally
geodesic submanifolds are obtained by intersecting the hyperbolic space H
n
with
Lorentzian subspaces.
If G acts irreducibly, then it m ust act tr ansitively on H
n
. By a previous observa-
tion, we already know that G is semisimple and of noncompact type. The second part
of the theorem follows from the theory of Riemannian symmetric spaces of noncom-
pact type, once we show that the isotropy group at some point is a maximal compact
subgroup of G.
By Theorem 2.6.2 there are no proper connected subgroups of SO
o
n,1
acting irre-
ducibly on R
n,1
. For signature 2 this is not the case. Di Scala and Leistner classified
in [106] the proper subgroups of SO
o
n,2
which act irreduc ibly on R
n,2
.
Theorem 2.6.5 (Di Scala, Leistner ) Let G be a proper connected Lie subgroup of
SO
o
n,2
which acts irreducibly on R
n,2
. Then G is conjugate to one of the following
subgroups:
(1) If n = 2m is even: U
m,1
,SU
m,1
or S
1
·SO
o
m,1
;
(2) If n = 3: A non-standard inclusion of SO
o
2,1
in SO
o
3,2
.
Submanifold Geometry of Orbits 61
2.7 Second fundamental form of orbits
Let G be a Lie group acting isometrically on a Riemannian manifold
¯
M.Let
p ∈
¯
M and A be the shape operator of the orbit M = G · p.Thenwehave
A
d
p
g(
ξ
)
d
p
g(X)=d
p
g(A
ξ
X)
for all g ∈ G, X ∈ T
p
M and
ξ
∈
ν
p
M. In particular, if M is a principal orbit and
ˆ
ξ
is
the equivariant normal vector field on M determined by
ξ
,then
A
ˆ
ξ
gp
d
p
g(X)=A
d
p
g(
ξ
)
d
p
g(X)=d
p
g(A
ξ
X)
for all g ∈ G, X ∈ T
p
M and
ξ
∈
ν
p
M. Therefore we have
A
ˆ
ξ
gp
= d
p
g ◦A
ξ
◦(d
p
g)
−1
, (2.4)
and hence:
Proposition 2.7.1 The principal curvatures of a principal orbit with respect to an
equivariant normal vector field are constant.
Let K = G
p
be the isotropy group at p and g = k ⊕m be a reductive decomposi-
tion. For X ∈ g we denote again by X
∗
the induced Killing vector field on
¯
M. Recall
that
T
p
M = {X
∗
p
: X ∈ g} = {X
∗
p
: X ∈ m}.
Let X ∈ g.SinceX
∗
is a Killing vector field on
¯
M,wehave(
¯
∇
X
∗
ξ
)
T
=(
¯
∇
ξ
X
∗
)
T
for
any normal vector field
ξ
on M,where(·)
T
denotes the orthogonal projection from
T
p
¯
M onto T
p
M. The Weingarten formula then tells us that
A
ξ
p
X
∗
p
= −((
¯
∇X
∗
)
p
ξ
p
)
T
.
Note that, since X
∗
is a Killing vector field on
¯
M, its covariant derivative (
¯
∇X
∗
)
p
at
p is a skewsymmetric endomorphism of T
p
¯
M. We summar ize this in:
Proposition 2.7.2 Let G be a Lie group acting isometrically on a Riemannian mani-
fold
¯
M. Then, for each p ∈
¯
M the tangent space of G · p at p is given by
T
p
(G ·p)={X
∗
p
: X ∈ g} = {X
∗
p
: X ∈ m},
where g = k ⊕m is a reductive decomposition of g and k is the Lie algebra of the
isotropy group K = G
p
at p. If
ξ
∈
ν
p
(G · p) and X ∈ g, the shape operator A
ξ
of
G · p at p with respect to
ξ
is given by
A
ξ
X
∗
p
= −((
¯
∇X
∗
)
p
ξ
)
T
,
where (·)
T
denotes the orthogonal projection from T
p
¯
M onto T
p
(G ·p).
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