300 Submanifolds and Holonomy
cept for
¯
M = G
2
2
/SO
4
and its compact dual space
¯
M = G
2
/SO
4
. It is an open problem
whether the equality i(
¯
M)=i
r
(
¯
M) holds for rk(
¯
M) ≥3. Using Proposition 11.1.9 one
can show that if
(i
r
(
¯
M) −1)i
r
(
¯
M) < 2(dim
¯
M −rk(
¯
M)),
then i(
¯
M)=i
r
(
¯
M). Using this estimate, and another refined one, Berndt and Olmos
verified the equality i(
¯
M)=i
r
(
¯
M) for several symmetric spaces. In the last column
of Table 11.4 we indicate whether the equality i(
¯
M)=i
r
(
¯
M) is known to hold or
whether it is still an open problem. The reflective index i
r
(
¯
M) itself can be calcu-
lated explicitly from Leung’s classification of reflective submanifolds in Riemannian
symmetric spaces (see [194–197]). We include i
r
(
¯
M) also in Table 11.4.
The relation between i(
¯
M) and i
r
(
¯
M) was studied in more detail by Berndt and
Olmos in [35]. As a consequence they obtained the classification of all irreducible
Riemannian symmetric spaces with i(
¯
M) ≤ 6.
11.2 Totally umbilical submanifolds and extrinsic spheres
11.2.1 Circles
We already discussed the existence and uniqueness of circles in Riemannian man-
ifolds in Proposition 10.4.3. It is well known that each geodesic in a Riemannian
symmetric space is an orbit of a one-parameter group of isometries. It is easy to
show that each circle in R
n
, S
n
, RP
n
and RH
n
is an orbit of a one-parameter group
of isometries. Maeda and Ohnita [203] proved that this is also true for circles in CP
n
and CH
n
. This was extended to all two-point homogeneous spaces by Mashimo and
Tojo [207]. In fact, they p roved that this property characterizes two-point homoge-
neous spaces.
Theorem 11.2.1 (Mashimo, Tojo) Let
¯
M be a Riemannian homogeneous space.
Then each circle in
¯
M is an o rbit of a one-parameter group of isometries if and
only if
¯
M is a two-point homogeneous space.
The “only if” part is proved by showing that the isotropy g roup at some point acts
transitively on unit tangent vectors at that point.
11.2.2 The classification problem for extrinsic spheres
One step toward the classification of extrinsic spheres of dimension ≥ 2inRie-
mannian symmetric spaces is the f ollowing result:
Submanifolds of Symmetric Spaces 301
Theorem 11.2.2 Let M be an extrinsic sphere in a Riemannian symmetric space
¯
M
with dimM ≥ 2. Then there exists a connected, complete, totally geodesic subma-
nifold N of
¯
M with constant curvature so that M is contained in N as an extrinsic
sphere with codimension one.
Proof Let M be an extrinsic sphere in a Riemannian symmetric space
¯
M and o ∈ M.
According to Theorem 10.4.4, M is uniquely determined by T
o
M and the mean cur-
vature vector H
o
by running along circles in
¯
M tangent to T
o
M and whose second
derivative at o is H
o
. As a consequence, we see that when there is a connected, com-
plete, totally geodesic submanifold N of
¯
M with o ∈ N and T
o
M ⊕RH
o
⊂ T
o
N,then
M ⊂ N. According to Corollary 11.1.1 we thus have to show that T
o
M ⊕RH
o
is a
curvature-invariant subspace of T
o
¯
M.
From Proposition 10.4.1 we already know that T
o
M is a curvature-invariant sub-
space of T
o
¯
M. If the codimension of M is one, we deduce from Theorem 11.1.6 that
¯
M has constant curvature. Since we treated this case in Theorem 1.6.2, we assume
from now on that the codimension of M is at least two.
Let
η
be a normal vector of M at o that is perpendicular to the mean curvature
vector H
o
of M at o. Then the shape operator A
η
of M with respect to
η
vanishes.
Further, since the mean curvature vector field H of M is parallel in the normal bundle
of M,wehaveR
⊥
(X,Y )H
o
= 0forallX,Y ∈ T
o
M,whereR
⊥
is the normal curvature
tensor of M. The Ricci equation therefore implies
¯
R(X,Y )H
o
,
η
= 0
for all X,Y ∈ T
o
M and all
η
∈
ν
o
M which are perpendicular to H
o
, and hence
¯
R(T
o
M,T
o
M)H
o
⊂ T
o
M ⊕RH
o
. (11.1)
So far, all arguments are true in the general case of a Riemannian manifold. We
will now use the assumption that
¯
M is a Riemannian symmetric space. In this case
the curvature tensor
¯
R is parallel, which means that
X
¯
R(Y, Z)W,U =
¯
R(
¯
∇
X
Y,Z)W,U+
¯
R(Y,
¯
∇
X
Z)W,U
+
¯
R(Y,Z)
¯
∇
X
W,U +
¯
R(Y, Z)W,
¯
∇
X
U (11.2)
for all tangent vector fields X,Y,Z,U,W on
¯
M.
We now assume that X,Y,Z,W are tangent to M and U =
η
is a normal vector
field o n M that is also perpendicular to H andsuchthat∇
⊥
η
vanishes at o.Then
¯
∇
X
η
vanishes at o. The left-hand side of equation (11.2) then vanishes because T
o
M
is a curvature-invariant subspace of T
o
¯
M. Using the equation
¯
∇
X
Y = ∇
X
Y + X,Y H,
similar equations for the other covariant derivatives, and (11.1), Equation (11.2) then
implies
0 = X ,Y
¯
R(H,Z)W,
η
+ X, Z
¯
R(Y, H)W,
η
.
302 Submanifolds and Holonomy
Since dim M ≥ 2, we can choose X = Y to be of unit length and Z perpendicular to
X. The previous equation then reduces to
0 =
¯
R(H,Z)W,
η
.
Since this holds for all Z,W (by varying with X)andall
η
, this implies
¯
R(H
o
,T
o
M)T
o
M ⊂ T
o
M ⊕RH
o
. (11.3)
We now choose X,Y,Z tangent to M, put W = H,andletU =
η
be a normal
vector field on M that is also perpendicular to H andsuchthat∇
⊥
η
vanishes at o.
The left-hand side of (11.2) vanishes because of Equation (11.1), which holds at each
point of M because o was chosen to be arbitrary. For the right-hand side, first observe
that
¯
∇
X
H = −A
H
X = −H,HX
and, once again,
¯
∇
X
η
= −A
η
X + ∇
⊥
X
η
= 0ato.
In a similar way, we then obtain
0 = X ,Y
¯
R(H,Z)H,
η
+ X, Z
¯
R(Y, H)H,
η
,
and conclude from this, using again the fact that dim M ≥ 2, that
¯
R(H
o
,T
o
M)H
o
⊂ T
o
M ⊕RH
o
. (11.4)
Eventually, Equations (11.1), (11.3), and (11.4), the algebraic curvature iden-
tities, and the fact that T
o
M is a curvature-invariant subspace of T
o
¯
M, im ply that
T
o
M ⊕RH
o
is a curvature-invariant subspace of T
o
¯
M. Thus, we have shown that
there exists a connected, complete, totally geodesic submanifold N of
¯
M so that M is
contained in N as an extrinsic sphere with codimension one. It remains to prove that
N has constant curvature.
To establish this, we first choose X = Y and Z = W to be orthonormal and tan-
gent and put U = H in Equation (11.2). The left-hand side vanishes because T
o
M is
curvature-invariant and o was chosen arbitrarily, and again using
¯
∇
X
H = −H,HX
we get
0 =
¯
R(H,Z)Z,H−H, H
¯
R(X,Z)Z,X.
If we denote by
¯
K(A, B) the sectional curvature of
¯
M with respect to the two-plane
spanned by A and B, this implies
0 = H,H(
¯
K(H, Z) −
¯
K(X,Z)).
Since H is nonzero everywhere, this implies
¯
K(X ,Z)=
¯
K(H, Z) (11.5)
whenever X and Z are orthonormal and tangent to M. An arbitrary 2-plane tangent to
N is spanned by orthonormal vectors of the form X and cos(
α
)Z + sin(
α
)
H
||H||
with
Submanifolds of Symmetric Spaces 303
X,Z orthonormal and tangent to M. A straightforward calculation, using Equation
(11.5) and once again the fact that T
o
M is curvature-invariant, yields
¯
R
X,cos(
α
)Z + sin(
α
)
H
||H||
cos(
α
)Z + sin(
α
)
H
||H||
,X =
¯
K(X ,H).
From this we conclude that, at each point of N, the sectional curvature in
¯
M of 2-
planes tangent to N is independent of the 2-plane. As N is totally geodesic in
¯
M,
we thus get that the sectional curvature of N depends only on the point. But N is
homogeneous, since it is a connected, complete, totally geodesic submanifold of a
symmetric space and hence itself a symmetric space. Thus, we eventually conclude
that N has constant sectional curvature.
Therefore, the classification of extrinsic spheres in Riemannian symmetric spaces
can be worked out in two steps. First, classify the totally geodesic submanifolds with
constant curvature in a Riemannian symmetric space. For symmetric spaces of com-
pact type, one can apply the results of Nagano and Sumi [228] mentioned in the
previous section. Using duality between symmetric spaces of compact and noncom-
pact type, the classification can be transferred to symmetric spaces of noncompact
type. In the second step, one has to classify the extrinsic spheres in spaces of constant
curvature with codimension one. This has been done explicitly in Theorem 1.6.2.
11.2.3 The classification problem for totally umbilical submanifolds
The classification of totally umbilical submanifolds of dimension > 2inRie-
mannian symmetric spaces has been achieved by Nikolaevskii [246]. Basically, these
submanifolds live in totally geodesically embedded products of spaces of constant
curvature. A partial classification was previously obtained by Chen in [79]. In partic-
ular, Chen proved:
Theorem 11.2.3 If an irreducible Riemannian symmetric space
¯
M contains a totally
umbilical hy persurface M, then both M and
¯
M have constant curvature.
A special case of this result is Theorem 11.1.6.
11.3 Symmetric submanifolds
11.3.1 Symmetry versus parallel second fundamental form
In Proposition 10.5.1 we proved that the second fundamental form of a symmet-
ric submanifold is parallel and that tangent to each normal space there exists a totally
geodesic submanifold of the ambient space. For simply connected Riemannian sym-
metric spaces, Naitoh [236] proved that the converse also holds.
304 Submanifolds and Holonomy
Proposition 11.3.1 (Naitoh) Let M be a complete submanifold of a simply con-
nected Riemannian symmetric space
¯
M. Then M is a symmetric submanifold if and
only if the second fundamental form of M is parallel and each normal space
ν
p
Mof
M is a curvature-invariant subspace of T
p
¯
M.
Proof First note that we have already given the proof of th is result in Theorem 2.8.2
in the special case when
¯
M is the Euclidean space. Suppose that the second funda-
mental form of M is parallel and that each normal space of M is curvature-invariant.
We fix a point p ∈ M and define a linear isometry
λ
on T
p
¯
M by
λ
: T
p
¯
M → T
p
¯
M , X →
−X if X ∈ T
p
M,
X if X ∈
ν
p
M.
Since the second fundamental form of M is parallel, the Codazzi equation implies
that T
p
M is a curvature-invariant subspace of T
p
¯
M. By assumption, the normal space
ν
p
M is also a curvature-invariant subspace of T
p
¯
M. The algebraic cu rvature identities
therefore imply
¯
R(T
p
M,T
p
M)T
p
M ⊂T
p
M,
¯
R(T
p
M,T
p
M)
ν
p
M ⊂
ν
p
M,
¯
R(T
p
M,
ν
p
M)T
p
M ⊂
ν
p
M,
¯
R(T
p
M,
ν
p
M)
ν
p
M ⊂ T
p
M, (11.6)
¯
R(
ν
p
M,
ν
p
M)T
p
M ⊂T
p
M,
¯
R(
ν
p
M,
ν
p
M)
ν
p
M ⊂
ν
p
M.
From this we easily derive that
λ
leaves
¯
R invariant, that is,
λ
(
¯
R(X,Y )Z)=
¯
R(
λ
(X),
λ
(Y ))
λ
(Z)
for all X,Y,Z ∈ T
p
¯
M. This implies (cf. [151]) that there exists a local isometry Λ of
¯
M with Λ(p)=p and whose differential at p coincides with the linear isometry
λ
.
However, since
¯
M is connected, complete, simply connected and real analytic, Λ can
be extended to a global isometry
σ
p
of
¯
M. By construction, we have
σ
p
(p)=p and d
p
σ
p
(X)=
−X if X ∈ T
p
M,
X if X ∈
ν
p
M.
AresultbyStr¨ubing [301] shows that there exists an open neighborhood U of p in M
such that
σ
p
(U) ⊂U. Completeness of M then eventually implies that
σ
p
(M)=M,
because s
p
reflects in p the geodesics in M through p.
11.3.2 Totally geodesic symmetric submanifolds
The classification of totally geodesic symmetric submanifolds in Riemannian
symmetric spaces follows from the one of reflective submanifolds (see Section
11.1.4).
Proposition 11.3.2 A totally geodesic submanifold M of a simply connected Rie-
mannian symmetric space
¯
M is symmetric if and only if it is a reflective submanifold.
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