Isoparametric Submanifolds and Their Focal Manifolds 165
We start with the first possibility. We can assume, by considering a neighborhood
of p if necessary, that all local orbits of Φ
¯p
in S(p) have the same dimension. Since
the integral manifolds S(x) move by parallel transport in the normal bundle of M
ξ
,
we can assume that the local orbit Φ
¯q
·q in S(q) is of maximal dimension for any
q ∈ M. In this way we obtain a distribution
˜
ν
on M given by the normal space
˜
ν
(q)
in S(q) of the orbit Φ
¯q
·q. Note that
˜
ν
(q)
⊥
is the tangent space of the holonomy
tube (M
ξ
)
−
ξ
(q)
⊂M at q. We will show that the distribution
˜
ν
satisfies the cond ition
of Moore’s Lemma 1.7.1, which then implies
˜
ν
= 0sinceM is locally irreducible
(
˜
ν
= TM,otherwise
ξ
would be umbilical) . The distribution
˜
ν
is autoparallel by
the Normal Holonomy Theorem and Proposition 2.3.11. By the proof of Proposition
2.3.11, if p
2
∈ L (p
1
),whereL (p
1
) denotes the integral manifold of
˜
ν
through p
1
,
then
η
p
1
= p
2
− p
1
belongs to the normal space of Φ
¯p
1
· p
1
at p
1
(regarded as a
submanifold of the affine normal space ¯p
1
+
ν
¯p
1
M
ξ
). If p
2
is close to p
1
then
η
p
1
is
fixed by the isotropy subgroup of Φ
¯p
1
at p
1
,sinceΦ
¯p
1
·p
1
and Φ
¯p
1
·p
2
have the same
dimension (note that ¯p
1
= ¯p
2
,sinceL (p
1
) ⊂ S(p
1
)). By Proposition 2 .3.5 (see also
Remark 2.3.10)
η
p
1
extends to a Φ
¯p
1
-invariant and ∇
⊥
-parallel normal vector field
on the orbit Φ
¯p
1
.p
1
in ¯p
1
+
ν
¯p
1
M
ξ
. It is now standard to show that
η
p
1
extends to a
parallel normal vector field
η
on the holonomy tube (M
ξ
)
−
ξ
(p
1
)
. Thus (M
ξ
)
−
ξ
(p
2
)
coincides with the parallel manifold ((M
ξ
)
−
ξ
(p
1
)
)
η
to (M
ξ
)
−
ξ
(p
1
)
.
Note that the distribution
˜
ν
consists of fixed points of the isotropy group of Φ
¯q
at q. Considering horizontal and vertical curves and applying similar arguments as in
Section 4.4.4, we obtain that
˜
ν
defines by restriction a ∇
⊥
-parallel flat subbundle of
the normal bundle of any holonomy tube (M
ξ
)
−
ξ
(q)
(regarded as a submanifold of the
ambient space). Moreover x +
η
(x) belongs to the leaf L (x) for any x ∈ (M
ξ
)
−
ξ
(q)
.
Let A, A
1
, A
2
and
¯
A be the shape operators of M, (M
ξ
)
−
ξ
(p
1
)
, (M
ξ
)
−
ξ
(p
2
)
and M
ξ
respectively. If H is the distribution given by the horizontal spaces of the holonomy
tubes (M
ξ
)
−
ξ
(q)
⊂ M (q ∈ M), then H (p
1
)=T
¯p
1
M
ξ
= H (p
2
), regarded as sub-
spaces of R
n
. Note that H = E
⊥
. The distribution H is invariant under the shape
operators of M and of the holonomy tubes. The restriction of
ξ
to any holonomy tube
(M
ξ
)
−
ξ
(q)
⊂ M, q ∈ M, is also a parallel normal vector field on this submanifold of
R
n
. This is a consequence of the invariance of H under shape the operators of M
and of the holonomy tube and the fact that A|
H
⊥
= id. Moreover, A
1
ξ
(p
1
)
and A
2
ξ
(p
2
)
have the same eigenvalues as A
ξ
(which are constant), with the possible exception of
1if
˜
ν
= ker(id −A
ξ
). This implies A
1
ξ
(p
1
)
= A
2
ξ
(p
2
)
if
η
is sufficiently close to 0. In
fact, both shape operators are simultaneously diagonalizable by the “tube formula”.
Note that
ξ
(p
i
)= ¯p
i
− p
i
, i = 1,2, so
ξ
(p
2
)=
ξ
(p
1
) −
η
(p
1
). By Lemma 4.5.5 we
have A
1
η
|
H
= 0, since H = E
⊥
=(ker(id −A
ξ
))
⊥
=(ker(id −A
1
ξ
))
⊥
(the distribu-
tions are restricted to (M
ξ
)
−
ξ
(p
1
)
). Thus
η
is constant along horizontal curves (with
respect to H ). But any two points in (M
ξ
)
−
ξ
(p
1
)
can be joined by a horizontal curve
(by the construction of the holonomy tube), so
η
is constant along (M
ξ
)
−
ξ
(p
1
)
in R
n
.
Therefore, the leaves of the autoparallel distribution
˜
ν
are parallel in R
n
along
any holonomy tube (M
ξ
)
−
ξ
(q)
.Since
˜
ν
⊥
is the distribution tangent to the holonomy
tubes,
α
(
˜
ν
,
˜
ν
⊥
)=0, where
α
is the second fundamental form of M.Furthermore,
˜
ν
is a parallel distribution on M, as it is autop a rallel and its leaves are parallel (in R
n
)
166 Submanifolds and Holonomy
along curves in M tangent to
˜
ν
⊥
. Moore’s Lemma then implies
˜
ν
= 0, since M is
locally irreducible.
We must still analyze the case when the orbit through p of the normal holonomy
group Φ
¯p
of M
ξ
at ¯p = p+
ξ
(p) in S(p) is not of local maximal dimension. This orbit
is always contained in S(p) near p (see the first part of the proof). Moreover, there
exist points p
∈S(p) arbitrary close to p such that the normal holonomy orbit Φ
¯p
.p
is of maximal dimension in S(p).Furthermore,Φ
¯p
.p
is a complete submanifold
of R
n
and coincides locally with S(p
) by what we have proved above. A standard
argument now shows that near p the orbit Φ
¯p
·p
locally coin cides with S(p),which
finishes the proof.
Proof of Theorem 4.5.2 Let
ξ
be a parallel normal isoparametric section on M.
Since M is contained in S
n
, we can assume that all eigenvalues
λ
i
of the shape op-
erator A
ξ
are different from zero (otherwise we can add a suitable constant multiple
of the position vector field to
ξ
). Let
ξ
i
=
λ
−1
i
ξ
and consider the focal manifold
M
ξ
i
= {p +
ξ
i
(p) : p ∈ M} and, for any p ∈ M, the holonomy tube (M
ξ
)
−
ξ
(p)
.By
Theorem 4.5.4, (M
ξ
)
−
ξ
(p)
coincides locally with M. The assertion now follows from
Remark 3.4.17, which states that, if all holonomy tubes (M
ξ
i
)
−
ξ
i
(p)
locally coincide
with M,thenM is a submanifold with constant principal curvatures.
A different proof of Theorem 4.5.2, using the Holonomy Lemma, can be found
in [94].
4.5.2 Global higher isoparametric rank rigidity
The global version of Theorem 4.5.2 is not trivial, since a simply connected
irreducible Riemannian manifold can be locally reducible at any point. The same
pathology probably exists in the context of submanifolds as well. The key fact for
this global version is the following result:
Lemma 4.5.6 Let M be a complete Riemannian manifold and let G act local polarly
on M. Let O be the open and dense subset of M consisting of all points p ∈ Mfor
which the G-orbit through p has maximal dimension. Assume that the distribution
ν
on O defined by the normal spaces of the orbits is (not only autoparallel but also)
parallel. Then O = M (that is, all orbits are maximal dimensional and
ν
defines a
parallel distribution on M).
Proof Let p ∈ M and v ∈
ν
p
(G ·p) be a principal vector for the slice representation
of G
p
. Then there exists
ε
> 0 such that G ·exp
p
(tv) is a principal orbit of G for all
t ∈ (0,
ε
) (see Exercise 2.11.4). Let X be a Killing vector field on M induced by G
p
with associated flow
φ
s
. Observe that
φ
s
(p)=p and that d
p
φ
s
(v) is a p rincipal vector
for the slice representation for all s ∈ R. Let us consider the map
f : R ×[0,
ε
) → M , (s,t) → f (s,t)=
φ
s
(
γ
v
(t)) =
γ
s
(t),
where
γ
s
=
γ
d
p
φ
s
(v)
.Define
ν
s,t
=
ν
γ
s
(t )
for t ∈(0,
ε
) and
ν
s,0
=(
τ
s
t
)
−1
ν
s,t
for t ∈(0,
ε
),
where
τ
s
t
denotes the parallel transport from 0 to t along the geodesic
γ
s
. Note that
Isoparametric Submanifolds and Their Focal Manifolds 167
ν
s,0
is well-defined since
ν
s,t
, t ∈ (0,
ε
), is parallel along f and in particular along
the geodesic
γ
v
. It is clear that
ν
s,t
varies smoothly along f . Moreover, being parallel
along the curve s → f (s,t) for t = 0,
ν
s,0
must be parallel along the constant curve
s → f (s,0)=p. This means that
ν
s,0
=
ν
0,0
for all s ∈ R.But
ν
s,0
is a slice of the
isotropy action that contains d
p
φ
s
(v) (see Exercise 2.11.6). Then d
p
φ
s
(v) ∈
ν
0,0
for
all s ∈ R.ThenX ·v =
d
ds
s=0
d
p
φ
s
(v) ∈
ν
0,0
.ButX ·v is perpendicular to
ν
0,0
,since
ν
0,0
is a slice for the isotropy action. Thus, X ·v = 0. Since X is an arbitrary Killing
vector field induced by G
p
, we conclude that the slice representation at p is discrete.
This means that G · p has maximal dimension for all p ∈ M.
For the global version of Theorem 4.5.2 we need the global version of Theorem
4.5.4. First of all we will make a few observations.
Let f : M → R
n
be a simply connected complete immersed submanifold and let
ξ
be a nonumbilical parallel normal isoparametric section on M. Assume that 1 is
an eigenvalue of the shape operator A
ξ
. Endow M with the “bundle-like” metric g
as in Exercise 4.6.6. Consider the quotient space M
/∼
,wherex ∼ y if x and y are
both in the same maximal integral manifold of ker(id −A
ξ
).ThenM
/∼
is endowed
with a natural differentiable Hausdorff manifold structure such that the projection
π
: M → M
/∼
is a smooth submersion. Moreover, if f
ξ
: M
/∼
→ R
n
is defined by
f
ξ
(
π
(p)) = f (p)+
ξ
(p),then f
ξ
is an immersion and M
/∼
with the induced metric is
a co mplete Riemannian m a nifold (Exercise 4.6.7) . With this procedure, startin g from
a complete submanifold M, we construct a parallel focal manifold f
ξ
: M
/∼
→ R
n
,
which is also complete with the induced metric.
As in Theorem 4.5.4, the (global) restricted normal holonomy group Φ
∗
π
(p)
of
M
/∼
acts on the fiber
π
−1
({
π
(p)}).Letc : [0,1] → M
/∼
be piecewise differentiable
with c(0)=c(1)=
π
(p) and let ˜c be its horizontal lift to M with ˜c(0)=p.As
in the local case, ˜c (t) −c(t) can be regarded as a parallel normal vector field on
f
ξ
: M
/∼
→ R
n
. As in the proof of Theorem 4.5.4, the normal spaces to maximal
dimensional orbits of Φ
∗
π
(p)
form a parallel distribution on
π
−1
({
π
(p)}). Applying
Lemma 4.5.6, we obtain that this parallel distribution is never singular. Using the
same ideas as for the proof of Theorem 4.5.4, but in the category of immersions, the
global version of Moore’s Lemma yields:
Theorem 4.5.7 ( [108]) Let f : M →R
n
be a full simply connected complete subma-
nifold and
ξ
be a nonumbilical parallel normal isoparametric section on M . Then,
for any p ∈ M, the holonomy tube (M
ξ
)
−
ξ
(p)
around the parallel focal manifold
f
ξ
: M
/∼
→ R
n
coincides with f : M → R
n
.
As in the local case we have a corollary, the global version of Theorem 4.5.2:
Theorem 4.5.8 ( [108]) Let f : M → S
n
be a full irreducible isometric immersion,
where M is a simply connected complete Riemannian manifold with iso-rank
f
(M) ≥
1. Then M is a submanifold with constant principal curvatures.
As an immediate consequence we see that if M is not an isoparametric hypersur-
face of S
n
, then it is an orbit of an s-representation.
168 Submanifolds and Holonomy
4.5.3 Higher isoparametric rank rigidity for s ubmanifolds of Euclidean
and hyperbolic spaces
We now investigate the case of submanifolds of Euclidean and hyperbolic spaces.
As a consequence of a Lorentzian version of Theorem 4.5.4, Olmos and Will proved
in [266] that the isoparametric rank of an irreducible full submanifold of a hyperbolic
space H
n
must be equal to zero (see Exercise 7.6.7).
Theorem 4.5.9 Let M be a submanifold of H
n
that is full and locally irreducible at
any point (regarded as a submanifold of R
n,1
). Then any parallel normal isopara-
metric section on M vanishes.
Using the same methods, regarding a submanifold M of R
n
as a submanifold of a
horosphere in H
n+1
⊂ R
n+1,1
, Olmos and Will proved in [266] the following result:
Theorem 4.5.10 Let M be a locally irreducible full submanifold of R
n
. If M admits
a nontrivial parallel normal isoparametric section
ξ
, then M is contained in a sphere
of R
n
.
As a consequence, if
ξ
is not a multip le of the radial vector field, then M has
constant principal curvatures by Theorem 4.5.2. In other words, Theorem 4.5.2 is
true for submanifolds of R
n
.Ifiso-rank
loc
(M) ≥ 1, then M is contained in a sphere,
and if iso-rank
loc
(M) ≥ 2, then M has constant principal curvatures.
Moreover, th e global versio n Theo rem 4.5.8 of Theorem 4.5.2 is true in the more
general context of submanifolds of R
n
( [108]). Namely, let f : M →R
n
be a full and
irreducible isometric immersion, where M is a simply connected complete Riemann-
ian man ifold. Then f (M) is contained in a sphere if iso-rank
f
(M) ≥ 1and f (M) is a
submanifold with constant principal curvatures if iso-rank
f
(M) ≥ 2.
By a result of Di Scala (Theorem 2.5.1; [105]), any minimal homogeneous sub-
manifold of R
n
is totally geodesic. Th e same is true f or H
n
by the results explained
in Section 2.6 [107]. So, we have the following result (cf. [258]):
Corollary 4.5.11 Let M be a homogeneous irreducible full submanifold (of positive
codimension) of H
n
or of R
n
which is not contained in a sphere. Then the mean
curvature vector field on M is not parallel.
4.6 Exercises
Exercise 4.6.1 Prove that if all leaves of a parallel foliation have parallel mean cur-
vature vector field, then each leaf has constant principal curvatures.
Isoparametric Submanifolds and Their Focal Manifolds 169
Exercise 4.6.2 Let D be an integrable distribution on an open subset of R
n
and as-
sume that the perpendicular distribution D
⊥
is integrable with totally g eodesic leaves
(or, equivalently, D
⊥
is autopar allel). The n a ny two (nearby) integral manifolds of
D are parallel. Equivalently, if M denotes one integral manifold, then the other one
is the parallel manif old M
ξ
with respect to some parallel normal vector field
ξ
on M.
In particular, any integral manifold has flat normal bundle.
Exercise 4.6.3 Let M be an isoparametric submanifold of S
n
and fix a unit parallel
normal vector field
ξ
on M in S
n
(more generally, let
ξ
be a parallel normal section).
Moving along the geodesic from any p ∈ M in direction
ξ
(p), consider the parallel
(possibly focal) manifolds
M
t
= {
ϕ
t
(p)=cos(t)p + sin(t)
ξ
(p) : p ∈ M}.
Prove the following “tube formula”: if
ξ
(p) is normal to M at p,then
¯
ξ
(
ϕ
t
(p)) =
−sin(t)p + cos(t)
ξ
(p) is normal to M
t
at
ϕ
t
(p) and the shape operator
¯
A of M
t
is
given by
¯
A
¯
ξ
=(sin(t)id + cos(t)A
ξ
)(cos(t)id −sin(t)A
ξ
)
−1
,
where we h ave to restrict to horizontal spaces in the case of a focal manifold.
Exercise 4.6.4 Let M be a submanifold of a space form
¯
M(
κ
) and assume that M is
contained in a totally umbilical submanifold N of
¯
M(
κ
). Prove that M is isoparamet-
ric in N if and only if it is isoparametric in
¯
M(
κ
).
Exercise 4.6.5 Prove that for a compact immersed full iso parametric submanifold
all curvature normals are nonzero.
Exercise 4.6.6 (cf. [108]) Let f : M → R
n
be an isometric immersion, where
(M,·,·) is a complete Riemannian manifold. Let
ξ
be an isopara metric parallel
normal vector field on M and
λ
= 0 an eigenvalue of the shape operator A
ξ
. Con-
sider the autoparallel distribution ker(id −A
λ
−1
ξ
) on M and define a Riemannian
metric g on M by requiring:
(i) ker(id −A
λ
−1
ξ
) and (ker(id −A
λ
−1
ξ
))
⊥
are perpendicular with respect to g.
(ii) g(X ,Y )=X,Y for all X,Y ∈ker(id −A
λ
−1
ξ
).
(iii) g(X,Y )=(id −A
λ
−1
ξ
)X, (id −A
λ
−1
ξ
)Y for all X,Y ∈ (ker(id −A
λ
−1
ξ
))
⊥
.
Prove that:
(a) (M,g) is a complete Riemannian manifold.
(b) Any maximal integral manifold of ker(id −A
λ
−1
ξ
) is an embedded closed sub-
manifold of M.
(c) Any two maximal integral manif olds S
1
and S
2
of ker(id −A
λ
−1
ξ
) are equidis-
tant with respect to g (that is, the distance d
g
(p,S
2
) does not depend on p ∈S
1
).
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