Chapter 5
Rank Rigidity of Submanifolds and Normal
Holonomy of Orbits
In the previous chapter we saw that orbits of s-representations agree, up to codimens-
ion two, with isoparametric submanifolds and their focal manifolds (or, equivalently,
with submanifolds with constant principal curvatures). It is therefore natural to look
for geometric invariants that distinguish orbits of s-representations from orbits of
other representations (or submanifolds with constant principal curvatures from other
submanifolds). In Chapters 3 and 4 we observed that the existence of a (nontrivial)
parallel normal isoparametric section strongly influences the geometry of a subma-
nifold.
In this chap ter we weaken this condition and require only that the submanifold
admits “sufficiently many” parallel normal vector fields, or in other words, that the
normal holonomy group has a nontrivial pointwise fixed subspace, whose d imension
is called the rank of the immersion. In the case of a homogeneous submanifold M of a
Euclidean space, Olmos proved in [257] that M is an orbit of an s-representation if the
rank is ≥2. In the original proof a crucial fact was that the curvature normals (defined
as in the isoparametric case, considering only directions in the flat part of the normal
bundle, as we will explain) of a homogeneous submanifold have constant length.
In [108] it is actually shown that this property, together with the same higher rank
assumption, yields a generalization (Theorem 5.1.7) of the above higher rank rigidity
result. In contrast to results on higher isoparametric rank rigidity (Theorems 4.5.2
and 4.5.8), Theorem 5 .1.7 is global and fails without the completeness assumption.
As a consequence we derive a global characterization of isoparametric submanifolds:
an immersed irredu cible complete submanifold f : M → R
n
with dimM ≥ 2andflat
normal bundle is isoparametric if and only if the distances to all focal hyperplanes
are constant on M.
In the last part of this chapter we apply these higher rank rigidity results to in-
vestigate the normal holonomy (and, more generally, ∇
⊥
-parallel transport) of ho-
mogeneous submanifolds. In the more general setting of homogeneous (pseudo)-
Riemannian vector bundles, the holonomy algebra can be described in terms of pro-
jection of Killing vector fields onto the homogeneous bundle (see [92] for more de-
tails). In the case of Riemannian manifolds this yields Kostant’s method for comput-
ing the Lie algebra of the holonomy group of a homogeneous Riemannian manifold.
Here we explain how to compute the normal holonomy of homogeneous submani-
folds by projecting the Killin g vector fields determined by the action on the normal
spaces (Theorem 5.2.7).
171
172 Submanifolds and Holonomy
5.1 Submanifolds with curvature normals of constant length and
rank of homogeneous submanifolds
In this section we investigate immersed submanifolds admitting p arallel normal
sections. In other words, we assume that the normal holonomy group has a nontrivial
pointwise fixed subspace, whose d imension is called the rank of the immersion. The
aim is to prove a global higher rank rigidity theorem, which is false in the lo cal
setting. In particular, this leads to a definition of isoparametricity that coincides with
the one used for complete submanifolds only. This condition can be formulated in
terms of the induced metric using the Gauss map, or equivalently, in terms of the
so-called third fundamental form.
5.1.1 Rank of submanifolds
Let f : M →
¯
M
n
(
κ
) be an immersed submanifold of a space form (with the in-
duced metric) and define the following subspaces of the normal space at p ∈ M:
(
ν
p
M)
0
= {
ξ
∈
ν
p
M : Φ
∗
p
·
ξ
=
ξ
},
(
ν
p
M)
s
=((
ν
p
M)
0
)
⊥
,
where Φ
∗
p
is the restricted normal holonomy group at p. Note that (
ν
M)
0
is the
maximal flat ∇
⊥
-parallel subbundle of
ν
M,where((
ν
M)
0
)
p
=(
ν
p
M)
0
.
Definition 5.1.1 The rank of the vector bundle (
ν
M)
0
over M is called the rank of
the submanifold M and is denoted by rank
f
(M). When there is no possible confusion,
we will write rank(M) instead of rank
f
(M).
If M is simply connected, then (
ν
M)
0
must be globally flat and so rank
f
(M) is the
maximal number of linearly independent parallel normal vector fields on M.Since
we are working in the category of immersions, we always assume that M is simply
connected. Otherwise, we consider the immersed submanifold f ◦
π
:
˜
M →
¯
M
n
(
κ
),
where
π
:
˜
M → M is the Riemannian universal covering space of M. In this situation
we have ((
ν
M)
0
)
π
(p)
=((
ν
˜
M)
0
)
p
for all p ∈
˜
M.
If M has flat normal bundle, then rank(M) is just the codimension of M in
¯
M
n
(
κ
).
Thus, if M is a full isoparametric submanifold of R
n
, the above notion of rank co-
incides with the notion of isoparametric rank. If, in addition, M is a homogeneous
submanifold, then it is a principal orbit of an s-representation by [96] and rank(M)
coincides with the rank of the corresponding symmetric space (see Section 4.4). Our
general philosophy is that in Euclidean submanifold geometry the normal holonomy
plays a role that is similar to the one of Riemannian holonomy in Riemannian geo-
metry, and orbits of s-representations play the r ole of Riemannian symmetric spaces.
By replacing the normal holonomy group with the local normal holonomy group,
we obtain the notion of the local rank rank
loc
f
(M)
p
at a point p ∈ M.Inotherwords,
Rank Rigidity of Submanifolds and Normal Holonomy of Orbits 173
rank
loc
f
(M)
p
is the maximal number o f linearly independent normal vector fields on
an open neighborhood of p.Thelocal rank of M is defined by
rank
loc
f
(M)=min{rank
loc
f
(M)
p
: p ∈ M}.
If M is simply connected, then rank
loc
f
(M)=rank
f
(M) if and o nly if any locally
defined parallel normal vector field extends to a p arallel normal vector field on M.
The previous equality holds in the two important cases when M is a real analytic
submanifold or when M has flat normal bundle.
Example 5.1.1 If M is an isoparametric hypersurface of the sphere S
n
and
M
1
,...,M
g
are its focal submanifolds, then
(
ν
M
i
)
0
= {0}
for all i ∈{1,...,g} (Exercise 5.3.1).
We will see that our notion of rank of a submanifold is particularly useful when M
is a homogeneous submanifold of R
n
that is contained in a sphere, in which case we
have rank(M) ≥ 1. This is the case for example when M is a compact homogeneous
submanifold of R
n
.
On the other hand, Will [343] discovered a family of homogeneous irreducible
full submanifolds of the real hyperbolic space H
n
with flat normal bundle and co-
dimension at least 2 (non-isoparametric by the classification of Section 4.2; see
also [348]). In this case, the rank does not interfere with the geometry of the ho-
mogeneous submanifold. We will therefore concentrate on (homogeneous) subman-
ifolds of Euclidean spaces (and spheres), where the existence of a nontrivial parallel
normal vector field has strong influence on the geometry. This is a special case of
the so-called submanifolds with curvature normals of constant length that we will
discuss next.
5.1.2 Submanifolds with curvature normals of constant length
Let f : M →R
n
be an immersed submanifold and assume that rank
f
(M) ≥1. Let
ξ
be a sectio n of (
ν
M)
0
.Since(
ν
M)
0
is ∇
⊥
-parallel and flat, we get R
⊥
(X,Y )
ξ
= 0
for all X,Y ∈ TM and so A
ξ
commutes with all shape operators of M by the Ricci
equation. In particular, A
ξ
commutes w ith all shape operators with respect to sec-
tions of (
ν
M)
0
. Therefore, for each p ∈ M,theset{A
ξ
:
ξ
∈ (
ν
p
M)
0
} is a family
of pairwise commuting selfadjoint endomorphisms. Simultaneous diagonalization of
the shape operators in this set induces an orthogonal decomposition
T
p
M = E
1
(p) ⊕...⊕E
g(p)
(p)
into distinct common eigenspaces. Associated with this decomposition are well-
defined normal vectors
η
i
(p) ∈ (
ν
p
M)
0
, called curvature normals, and linear forms
λ
i
(p) : (
ν
p
M)
0
→ R such that
A
ξ
|
E
i
(p)
=
λ
i
(p)(
ξ
)id
E
i
(p)
=
η
i
(p),
ξ
id
E
i
(p)
174 Submanifolds and Holonomy
for all
ξ
∈ (
ν
p
M)
0
and i ∈{1,...,g(p)}. For each E
i
(p) we denote by E
⊥
i
(p) the
orthogonal complement of E
i
(p) in T
p
M. The dimension of E
i
(p) is called the mul-
tiplicity of the curvature normal
η
i
(p). This corresponds to the multiplicity of the
eigenvalue
λ
i
(p)(
ξ
) of the shape operator A
ξ
for a generic normal vector
ξ
∈(
ν
p
M)
0
(generic means here that
ξ
is not in the union of the hyperplanes orthogonal to
η
i
(p) −
η
j
(p), i, j ∈{1,...,g (p)}, i = j) . Sometimes it is convenient to consider
curvature normals at p as an m-tuple (
η
1
(p),...,
η
m
(p)),wherem = dimM and each
curvature normal is counted with multiplicity.
The curvature normals have the usual continuity property. The proof of the fol-
lowing proposition is straightforward.
Proposition 5.1.2 (Continuity property of curvature normals) Let (p
k
)
k∈N
be a
sequence of points p
k
∈ M converging to p ∈ M and let (
η
1
(p
k
),...,
η
m
(p
k
)) be the
curvature normals at p
k
(chosen in any order and counted with multiplicities). Then
there exists a subsequence (p
k
j
)
j∈N
such that (
η
1
(p
k
j
),...,
η
m
(p
k
j
)) converges to
the curvature normals (
η
1
(p),...,
η
m
(p)) at p (order is not necessarily preserved).
There exists an open and dense subset Ω of M on which the number g(p) of dis-
tinct eigenspaces is locally constant, o r equivalently, the number of d istinct curvature
normals is locally constant. On Ω the eigenspaces locally define smooth distributions
and the corresponding curvature normals are locally defined smooth normal sections.
It is standard to show, using the Codazzi equation, that on Ω each distribution E
i
is
integrable (in general, the leaves are not totally umbilical unless (
ν
M)
0
=
ν
M;see
Lemma 3.4.2). If rk(E
i
) ≥ 2, then ∇
⊥
X
η
i
= 0forallX tangent to E
i
by the Codazzi
equation. The ∇
⊥
-parallelism of
η
i
in directions orthogonal to E
i
is equivalent to the
autoparallelism of E
i
. Once again, the main ingredient is the Codazzi equation.
Lemma 5.1.3 Let f : M → R
n
be an immersed submanifold with rank
f
(M) ≥ 1.Let
U be a connected open subset of M on which the common eigenspaces define smooth
distributions E
1
,...,E
g
.
(a) The distribution E
i
is autoparallel if and only if ∇
⊥
X
η
i
= 0 for all sections X in
E
⊥
i
.
(b) If rk(E
i
) ≥ 2,thenE
i
is autoparallel if and only if
η
i
is parallel.
Proof Let X ,Y, Z
j
be tangent vector fields on M such that X and Y are tangent to E
i
and Z
j
is tangent to E
j
for j = i.Let
ξ
be a generic parallel section of (
ν
U)
0
.Then
we have
(∇
X
A)
ξ
Y,Z
j
=(
λ
i
(
ξ
) −
λ
j
(
ξ
))∇
X
Y,Z
j
.
Thus we have (∇
X
A)
ξ
Y,Z = 0 for all sections Z in E
⊥
i
if and only if ∇
X
Y is a
section in E
i
, which is equivalent to E
i
being autoparallel. Using the Codazzi equation
we get
(∇
X
A)
ξ
Y,Z = (∇
Z
A)
ξ
X,Y = d
λ
i
(
ξ
)(Z)X,Y = ∇
⊥
Z
η
i
,
ξ
X,Y .
Since generic normal vectors form an open and dense subset of a normal space, we
Rank Rigidity of Submanifolds and Normal Holonomy of Orbits 175
getpart(a).Ifrk(E
i
) ≥ 2 we can choose orthonormal sections X and Y in E
i
. Then,
from the Codazzi equation, ∇
⊥
X
η
i
= 0andso∇
⊥
E
i
η
i
= 0. Now (b) follows from (a).
Remark 5.1.4 Observe that R
⊥
(X,Y )
ξ
= 0 for all sections
ξ
∈ (
ν
M)
0
.Fromthe
Ricci equation we have R
⊥
(X,Y )
ξ
,
η
= [A
ξ
,A
η
]X,Y for all
η
∈
ν
M,soA
ξ
com-
mutes with all shape operators. Any curvature distribution E
i
is then invariant un-
der all shape operators, or equivalently, the second fundamental form
α
satisfies
α
(E
i
,E
j
)=0foralli = j.
Definition 5.1.5 An immersed submanifold f : M → R
n
is said to have curvature
normals of constant length if the set
L
p
= {
η
1
(p),...,
η
g(p)
(p)}⊂ R
does not depend on p ∈ M.
It is interesting to note that an extrinsically homogeneous submanifold M of R
n
with rank(M) ≥ 1 has curvature normals of constant length. In fact, if h ∈ I(R
n
)
satisfies h(M)=M and h(p)=q,thend
p
h(
ν
p
M)=
ν
q
M and the linear isometry
d
p
h|
ν
p
M
:
ν
p
M →
ν
q
M maps curvature normals at p to curvature normals at q.
For each p ∈ M we define
L
2
p
= {
η
1
(p)
2
,...,||
η
g(p)
(p)
2
}⊂R.
Then M has curvature normals of constant length if and only if L
2
p
does not depend
on p ∈ M. An interesting fact is that L
2
p
is related to the eigenvalues of the so-called
adapted third fundamental form, which we will introduce next. Let k = rank
f
(M)
and denote by G(k,n) the Grassmann manifold of k-planes in R
n
.Themap
G
0
: M → G(k,n) , p → (
ν
p
M)
0
is called the adapted Gauss map of M.If
ν
M is flat, then the adapted Gauss map
coincides with the usual Gauss map of M.Letg
0
be the possibly degenerate metric
on M that is induced by the adapted Gauss map. The symmetric tensor field B
0
on M
defined by
g
0
(X,Y )=B
0
X,Y
is called the adapted third fundamental form of M.If
ν
M is flat, then B
0
coincides
with the third fundamental form on M, which is classically defined by means of the
usual Gauss map (see [250]). If
ξ
1
,...,
ξ
k
is an orthonormal basis of (
ν
p
M)
0
then
(see Exercise 5.3.4)
B
0
p
=
k
∑
j=1
A
2
ξ
j
. (5.1)
Since A
ξ
j
|
E
i
(p)
=
η
i
(p),
ξ
j
id
E
i
(p)
, equation (5.1) yields
B
0
p
E
i
(p)
=
k
∑
j=1
η
i
(p),
ξ
j
2
id
E
i
(p)
= ||
η
i
(p)||
2
id
E
i
(p)
for all i ∈{1,...,g(p)}. Thus, we have proved:
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