Isoparametric Submanifolds and Their Focal Manifolds 145
other words, the Coxeter group of an isoparametric hypersurface in S
n
is cry stallo-
graphic, that is, it stabilizes a lattice in R
2
(see [156, Section 2.8]), see also [275].
The fact that W is a finite reflection group implies that the intersection of the focal
hyperplanes
i
(p) is nonempty and hence W has a fixed point. This implies that an
isoparametric submanifold of R
n
with rank ≥ 1 admits a parallel normal vector field
ζ
on M such that
ζ
,n
i
= 1foralli ∈{1,...,g}. This observation will be crucial
for the next reduction results.
4.2.3 Reduction theorems for isoparametric submanifolds of Euclidean
space
The Coxeter group allows us to prove reduction results for isoparametric sub-
manifolds. We begin with the Euclidean setting. Roughly speaking, what happens is
that, given an isoparametric submanifold of a Euclidean space, we can always split
off its Euclidean factor. The remainder is a product of compact isoparametric sub-
manifolds with an irreducible Coxeter group. Our first result states that the Coxeter
group has a fixed point.
Proposition 4.2.9 Let M be a connected complete isoparametric submanifold of R
n
such that all curvature normals do not vanish (e.g., if M is compact). Then M is
contained in a sphere.
Proof Let
ζ
be a parallel normal vector field on M such that
ζ
,n
i
= 1forall
i ∈{1,...,g} (its existence follows from the existence of a fixed point for the Weyl
group of M). Then f : M →R
n
, p → p +
ζ
p
is a constant map. Indeed, for any tangent
vector field X on M we have d
p
f (X)=X −A
ζ
X = 0sinceTM = E
1
⊕...⊕E
g
and
A
ζ
X
i
= n
i
,
ζ
X
i
= X
i
for all sections X
i
in E
i
.SinceM is connected it follows that f
is constant, and since
ζ
has constant length it follows that M lies on the sphere with
radius
ζ
and center p +
ζ
p
with p ∈ M arbitrary.
As an immediate consequence we g et the following nice characterization of com-
pact isoparametric submanifolds of Euclidean spaces.
Corollary 4.2.10 Let M be a connected complete isoparametric submanifold of R
n
.
Then the following statements are equivalent:
(i) M is compact.
(ii) All curvature normals of M are nonzero.
(iii) M is contained in a sphere in R
n
.
More generally, if M is noncompact, then we can split off a Euclidean factor and
M is locally an extrinsic p roduct of an isoparametric submanifold of a sphere with a
Euclidean factor.
146 Submanifolds and Holonomy
Theorem 4.2.11 (Palais, Terng [275]) Let M be a complete isoparametric subma-
nifold of R
n
with rank k and assume that zero is a curvature normal of M (or equiva-
lently, that the nullity distribution E
0
on M is nontrivial). Then there exists a compact
isoparametric submanifold M
1
of a sphere in R
n−m
0
such that M splits as the extrin-
sic direct product of M
1
and a leaf of E
0
.
Proof Let
ζ
be a parallel normal vector field on M such that n
i
,
ζ
= 1forall
i ∈{1,...,g}.Then
ζ
focalizes each of the curvature distribution E
i
, i ∈{1,...,g}.
In other words, ker(id −A
ζ
)=E
1
⊕...⊕E
g
= E
⊥
0
.SinceE
0
and ker(id −A
ζ
) are
both autoparallel distributions on M that are invariant under all shape operators of M,
Moore’s Lemma 1.7.1 implies that M splits as stated.
A local version of the theorem is straightforward.
There is another useful splitting result for isoparametric submanifolds. Suppose
that M
1
and M
2
are isoparametric submanifolds of R
m
1
+k
1
and R
m
2
+k
2
, respectively.
Let W
i
be the Coxeter group of M
i
.ThenM
1
×M
2
is an isoparametric submanifold
of R
m
1
+m
2
+k
1
+k
2
with Coxeter group W
1
×W
2
. The converse is also true.
Theorem 4.2.12 Let M be an isoparametric submanifold of R
n
without Euclidean
factor (or equivalently, M is contained in a sphere). Then M is reducible if and only
if its Coxeter group W is reducible.
Proof We have only to show that if W splits as W
1
×W
2
,whereW
i
is a Coxeter group
on R
k
i
,thenM splits as an extrinsic product of two isoparametric submanifolds M
1
and M
2
. The converse is clear.
By assumption, all curvature normals n
1
,...,n
g
of M are nonzero. Let
ζ
be a
parallel normal vector field on M such that n
i
,
ζ
= 1foralli ∈{1,...,g}.SinceW
splits into W
1
×W
2
, this induces a decomposition
ζ
=
ζ
1
+
ζ
2
so that
ζ
1
(resp.
ζ
2
)
is perpendicular to the curvature normals for W
2
(resp. W
1
). The distributions D
1
=
ker(id −A
ζ
1
) and D
2
= ker(id −A
ζ
2
) on M are mutually orthogonal, autoparallel and
invariant b y all shape operators of M. Hence both distributions are parallel and the
statement follows from Moore’s Lemma 1.7.1.
4.2.4 The Slice Theorem
Using essentially the same arguments as in the proof for Lemma 3.4.2, we have
the following:
Lemma 4.2.13 Let M be a complete isoparametric submanifold of R
n
and
ξ
be a
parallel normal vector field o n M such that M
ξ
is a parallel or focal manifold of M.
Let
π
: M →M
ξ
be the focal map and q ∈M
ξ
. Then every connected component of the
fiber
π
−1
({q}) is a compact isoparametric submanifold of the normal space
ν
q
M
ξ
of M
ξ
at q. Moreover, if
ξ
satisfies n
i
,
ξ
= 1 for the curvature normals n
1
,...,n
g
of
M, then the restrictions n
1
|
π
−1
({q})
,...,n
g
|
π
−1
({q})
are the curvature normals of the
isoparametric submanifold
π
−1
({q}) of
ν
q
M
ξ
.
Isoparametric Submanifolds and Their Focal Manifolds 147
Recall that, for a fixed isoparametric submanifold M of R
n
, the parallel and focal
manifolds of M determine a singular foliation of R
n
. The isoparametric leaves in that
foliation correspond to normal vectors in the interior of a Weyl chamber of the Weyl
group at a point p ∈ M. From the previous lemma we can deduce the important Slice
Theorem of Hsiang, Palais, and Terng [155].
Theorem 4.2.14 (Slice Theorem) Let M
ξ
be a focal manifold of a complete
isoparametric submanifold M of R
n
.Letq∈ M
ξ
and V =
ν
q
M
ξ
. Then the paral-
lel manifolds of M intersect V in an isoparametric foliation of V .
We now compare the normal spaces of parallel focal manifolds and prove a gen-
eralization of Proposition 3.4.11 for isoparametric submanifolds.
Lemma 4.2.15 Let M be a complete isoparametric submanifold of R
n
and M
ξ
be a
parallel or focal manifold of M. Let
π
: M → M
ξ
be the parallel or focal map and
q ∈ M
ξ
.Then
q +
ν
q
M
ξ
=
!
p∈
π
−1
({q})
(p +
ν
p
M).
Proof From Lemma 4.2.13 we know that each connected component of
π
−1
({q}) is
an isoparametric submanifold of
ν
q
M
ξ
. Thus
ν
q
M
ξ
is the union of the leaves of the
singular foliation d etermined by
π
−1
({q}).Let
η
∈
ν
q
M
ξ
and choose p ∈
π
−1
({q})
such that the Euclidean distance from
η
to th e compact set
π
−1
({q}) attains its min-
imum at p.Then
η
− p must be perpendicular to T
p
π
−1
({q}). The vector
η
− p is
also perpendicular to M at p,since
η
−p ∈
ν
q
M
ξ
and T
q
M
ξ
=(T
p
π
−1
({q}))
⊥
∩T
p
M.
Thus
η
belongs to the affine normal space p +
ν
p
M. This shows that the right-hand
term of the equality contains the left-hand term. The other inclusion is clear since
q +
ν
q
M
ξ
contains any term of the union in the right-hand side o f the equation.
4.2.5 Applications to isoparametric hypersurfaces of spheres
In this subsection we will investigate the case o f isoparametric hypersurfaces in
spheres more closely. We can use Coxeter groups to derive an explicit formula for
the principal curvatures. We will also obtain a formula for the principal curvatures of
the focal manifolds, which implies that they are always minimal submanifolds. This
will also yield an alternative proof for Cartan’s fundamental formula 2.9.4.
Let M be a connected isoparametric hypersurface of the unit sphere S
n
⊂ R
n+1
,
p ∈ M and
ξ
∈
ν
p
M ∩T
p
S
n
be a unit normal vector of M at p so that (
ξ
,−p) is
an orthonormal frame for the 2-dimensional normal space
ν
p
M of M at p in R
n+1
.
Every unit normal vector
η
∈
ν
p
M can be written in the form
η
= cos(t)
ξ
+ sin(t)(−p)=cos(t)
ξ
−sin(t)p.
If
λ
1
,...,
λ
g
are the distinct principal curvatures of M with respect to
ξ
, then the prin-
cipal curvatures of M with respect to
η
are cos(t)
λ
1
+ sin(t),...,cos(t)
λ
g
+ sin(t).It
follows that
n
1
(p)=
λ
1
ξ
− p,...,n
g
(p)=
λ
g
ξ
− p
148 Submanifolds and Holonomy
are the curvature normals of M at p. We label the principal curvatures so that
λ
i
= cot(
θ
i
) with 0 <
θ
1
<...<
θ
g
<
π
.
Geometrically,
θ
i
is the angle between
ξ
and n
i
(p). From properties of Coxeter
groups with rank 2 (see Example 4.2.1) we obtain
θ
i
−
θ
i−1
=
π
g
.
We thus have proved the following
Theorem 4.2.16 (M¨unzner) Let M be an isoparametric hypersurface of S
n
with dis-
tinct principal curvatures
λ
i
= cot(
θ
i
) with 0 <
θ
1
<...<
θ
g
<
π
and multiplicities
m
i
.Then
θ
k
=
θ
1
+
k −1
g
π
for all k ∈{1,...,g}
and the multiplicities satisfy m
i
= m
i+2
(index modulo g).
Next, we consider parallel and focal manifolds of M in S
n
.Let
ξ
be a unit normal
vector field on M in S
n
. The geodesic
γ
in S
n
with
γ
(0)=p and
˙
γ
(0)=
ξ
p
is given
by
γ
(t)=cos(t)p +sin(t)
ξ
p
. It follows that the parallel or focal manifold M
t
of M at
distance t is
M
t
= {cos(t)p + sin(t)
ξ
p
: p ∈ M}.
We d efine the smooth map
ϕ
t
: M → M
t
, p → cos(t)p + sin(t)
ξ
p
.
The differential of
ϕ
t
at p is
d
p
ϕ
t
= cos(t)id −sin(t)A
ξ
.
Let TM = E
1
⊕...⊕E
g
be the orthogonal decomposition of TM into the curvature
distributions E
1
,...,E
g
.ForX
i
∈ E
i
(p) we then get
d
p
ϕ
t
(X
i
)=(cos(t) −sin(t) cot(
θ
i
))X
i
=
sin(
θ
i
−t)
sin(
θ
i
)
X
i
.
It follows that
ϕ
t
(p) is a focal point of M if and only if t ∈
θ
i
+
π
Z for some i ∈
{1,...,g}. Thus, the focal manifold M
i
= M
θ
i
focalizes the curvature distribution E
i
on M.
As in the case of submanifolds of Euclidean space (cf. Lemma 3.4.7), there exists
a “tube formula” for the shape operator of M
t
. The geodesic
γ
intersects the parallel
and focal manifolds of M perpendicularly at each point, so
ξ
t
p
= −sin(t)p + cos(t)
ξ
p
Isoparametric Submanifolds and Their Focal Manifolds 149
is a unit normal vector of M
t
at
ϕ
t
(p). The shape operator A
t
ξ
t
p
of M
t
with respect to
ξ
t
p
is given by
A
t
ξ
t
p
=(sin(t)id + cos(t)A
ξ
)(cos(t)id −sin(t)A
ξ
)
−1
, (4.5)
where we have to restrict to horizontal spaces in the case of focal manifolds (Exercise
4.6.3).
A simple calculation shows that for a parallel manifold M
t
the principal curva-
tures with respect to
ξ
t
of M
t
are
cot(
θ
1
−t),...,cot(
θ
g
−t)
(see also [70], page 246). For a focal manifold M
i
= M
θ
i
the principal curvatures with
respect to
ξ
i
p
=
ξ
θ
i
p
at
ϕ
i
(p)=
ϕ
θ
i
(p) are
cot(
θ
k
−
θ
i
)=
1 +
λ
i
λ
k
λ
i
−
λ
k
, k =∈{1,...,g} , k = i.
A straightforward generalization of Lemma 4.2.15 shows that the normal vectors
ξ
i
p
with p ∈
ϕ
−1
i
({q}) generate the normal space of the focal manifold M
i
at q ∈ M
i
.In
particular, we see that the principal curvatures of the focal manifold are independent
of the (unit) normal vector.
We now apply Theorem 4. 2.16 to prove the minimality of focal manifolds.
Corollary 4.2.17 Each focal manifold M
i
of an isoparametric hypersurface M of S
n
is a minimal submanifold of S
n
.
Proof Let
ξ
i
p
be a unit normal vector of M
i
at
ϕ
i
(p) and A
i
ξ
i
p
be the shape operator of
M
i
with respect to
ξ
i
p
.Then
tr
A
i
ξ
i
p
=
∑
k=i
m
k
cot(
θ
k
−
θ
i
)=
g−1
∑
k=1
m
k
cot
k
π
g
= −
g−1
∑
k=1
m
k
cot
π
−
k
π
g
= −
g−1
∑
k=1
m
k
cot
(g −k)
π
g
= −
g−1
∑
k=1
m
g−k
cot
k
π
g
.
From Theorem 4.2.16 we know that m
g−k
= m
k
. So the last term in the previous equa-
tion is equal to −tr
A
i
ξ
i
p
and therefore tr
A
i
ξ
i
p
= 0. Since the principal curvatures
of M
i
are independent of the (unit) normal vector, it follows that M
i
is a minimal
submanifold.
Since cot(
θ
j
−
θ
i
)=
1+
λ
i
λ
j
λ
i
−
λ
j
,wealsohave
tr
A
i
ξ
i
p
=
∑
k=i
m
k
1 +
λ
i
λ
k
λ
i
−
λ
k
.
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