140 Submanifolds and Holonomy
{A
ξ
p
}
ξ
p
∈
ν
p
M
have a common basis of eigenvectors. Then there exist an orthogonal
decomposition
T
p
M = E
0
(p) ⊕E
1
(p) ⊕...⊕E
g
(p)
of T
p
M and g distinct nonzero linear forms
λ
1
(p),...,
λ
g
(p) on
ν
p
M such that
A
ξ
p
X
i
=
λ
i
(p)(
ξ
p
)X
i
for all X
i
∈ E
i
(p) , i ∈{0,...,g},
where
λ
0
(p) is the zero linear form on
ν
p
M. Note that g = 0 is possible, for example
for R
n−1
⊂ R
n
,andalsoE
0
(p)={0} is possible, for example for S
n−1
⊂ R
n
.For
i ∈{0,...,g} the metric dual vector n
i
(p) ∈
ν
p
M of
λ
i
(p),givenby
λ
i
(p)(
ξ
p
)=
n
i
(p),
ξ
p
for all
ξ
p
∈
ν
p
M, is called a curvature normal of M at p. We obviously
have n
0
(p)=0andn
i
(p) = 0foralli ∈{1,...,g}.
Since M is isoparametric, these objects at p can be extended smoothly to objects
on M. This gives an orthogonal decomposition
TM = E
0
⊕E
1
⊕...⊕E
g
of the tangent bundle TM into smooth distributions E
0
,E
1
,...,E
g
consisting of eigen-
vectors of all shape operators. The rank m
i
of E
i
is called a multiplicity of M.The
linear forms extend to real-valued one-forms
λ
i
on
ν
M so that
λ
i
(
ξ
) give the eigen-
values of the shape operator A
ξ
for all normal vector fields
ξ
on M. The curvature
normals extend to global normal vector fields n
i
on M. Since the eigenvalues
λ
i
(
ξ
)
are constant if
ξ
is a parallel normal vector field, the curvature normals n
i
are parallel
normal vector fields on M .
Remark 4.2.1 If M is an isoparametric submanifold of H
n
, the curvature normals are
well-defined since M is a Riemannian submanifold of R
n,1
. We can also define cur-
vature normals ˜n
i
by regarding M as a submanifold of H
n
.Thenwehaven
i
= ˜n
i
+ p,
where p is the position vector field. In particular, n
i
= 0foralli ∈{1,...,g}.More-
over, the curvature distributions associated with n
i
and ˜n
i
coincide as the position
vector field is u mbilical.
The decomposition TM = E
0
⊕E
1
⊕...⊕E
g
can be realized by a single shape
operator as follows. Let p ∈ M and choose
ξ
p
∈
ν
p
M so that
ξ
p
is not contained
in any of the hyperplanes perpendicular to the vectors n
i
(p) and n
i
(p) − n
j
(p),
i, j ∈{1,...,g}, i = j.Let
ξ
be the parallel normal vector field on M whose value at
p is the given one. Then the eigenspaces of the shape operator A
ξ
form the decompo-
sition TM = E
0
⊕E
1
⊕...⊕E
g
.Since
ξ
is a parallel normal isoparametric section, it
follows from Lemma 3.4.2 that the distributions E
0
,E
1
,...,E
g
on M are autoparallel
and hence induce totally geodesic foliations of M. We denote by S
i
(p) the maximal
leaf containing p of the autoparallel distribution E
i
on M. The leaf S
i
(p) is called a
curvature leaf of M an d its dimension is equal to the multiplicity m
i
. Each subbundle
E
i
is invariant under all shape operators of M, which operate on E
i
as multiples of
the identity. It follows that S
i
(p) is a totally umbilical subman ifold of the Euclidean
or Lorentzian space and contained in the affine subspace p +(E
i
(p)+
ν
p
M).Inthe
Euclidean case, S
0
(p) is an open part of an affine subspace and S
i
(p), i ∈{1,...,g},
Isoparametric Submanifolds and Their Focal Manifolds 141
is an open part of a sphere. In the Lorentzian case, S
0
(p) is an open part of an affine
subspace and S
i
(p), i ∈{1,...,g}, is an open part of a totally umbilical hyperbolic
subspace if n
i
(p) < 1, of a horosphere if n
i
(p) = 1, or of a totally umbilical
sphere if n
i
(p) > 1.
Lemma 4.2.2 Let M be an isoparametric submanifold of R
n
or H
n
(regarded as a
Riemannian submanifold of R
n,1
). Then M is full if and only if the curvature normals
n
1
(p),...,n
g
(p) span the normal space
ν
p
M a t some (and hence any) point p ∈ M.
In particular, if M is full, then the inequality codimM ≤ dimM holds.
Proof Since the curvature normals are parallel normal vector fields, it is clear that
n
1
(p),...,n
g
(p) span
ν
p
M at p ∈ M if and only if n
1
(q),...,n
g
(q) span
ν
q
M at any
point q ∈ M. The curvature normals n
1
(p),...,n
g
(p) do not span
ν
p
M if and only
if there exists a nonzero parallel normal vector field
ξ
on M that is perpendicular to
all curvature normals n
i
.SinceA
ξ
= 0, this is equivalent to M not being full by The-
orem 1.5.1 on the reduction of the codimension (which also applies to Riemannian
submanifolds of Lorentzian spaces).
Remark 4.2.3 In view of Lemma 3.1.3, the previous lemma can be reformulated as
follows: the first normal space of an isoparametric submanifold is spanned by the
curvature normals.
Using Lemma 4.2.2 we obtain that for i ∈{1,...,g} the curvature leaf S
i
(p) is
contained in the affine space p +(E
i
(p)+Rn
i
(p)). This is due to Lemma 3.4.2 and
the fact that the first normal space is spanned by the n
i
. So, in the Euclidean case, we
have
Lemma 4.2.4 Let M be an isoparametric submanifold of R
n
and p ∈ M.
(i) For each i ∈{1,...,g}the c urvature leaf S
i
(p) is an open part of a sphere with
center p in the affine subspace p +(E
i
(p)+Rn
i
(p)).
(ii) The curvature leaf S
0
(p) is an open part of the affine subspace p + E
i
(p) .
Note that f or an immersed full compact isoparametric submanifold the distribu-
tion E
0
must be trivial since any leaf S
0
(p) is an affine subspace (as M is compact
and hence complete) and hence must be trivial since M is compact.
Let M be an isoparametric submanifold of a Euclidean or Lorentzian space and
ξ
be a parallel normal vector field on M.Then
ξ
is a parallel normal isoparametric
section and as in Section 3.4 we can consider the parallel or focal manifold M
ξ
.
Recall that p +
ξ
p
is a focal point of M if and only if ker(id −A
ξ
p
) = 0. We can
write down this condition taking into account the curvature normals. Since A
ξ
|
E
i
=
λ
i
(
ξ
)id
E
i
= n
i
,
ξ
id
E
i
for i > 0andA
ξ
|
E
0
= 0, we get the matrix
id −A
ξ
=
⎛
⎜
⎜
⎜
⎝
10··· 0
01−n
1
,
ξ
··· 0
.
.
.
.
.
.
.
.
.
.
.
.
00··· 1 −n
g
,
ξ
⎞
⎟
⎟
⎟
⎠
142 Submanifolds and Holonomy
with respect to the decomposition TM = E
0
⊕E
1
⊕...⊕E
g
.ThesetF
M
(p) of focal
points of M at p therefore is the union over i ∈{1,...,g} of the hyperplanes
i
(p)={p +
ξ
p
: n
i
(p),
ξ
p
= 1}.
The hyperplane
i
(p) is called the focal hyperplane associated with the eigenspace
E
i
(p).
Remark 4.2.5 The focal hyperplanes of an isoparametric submanifold are invariant
under parallel transport in the normal bundle, since the curvature normals are ∇
⊥
-
parallel.
It follows that an isoparametric submanifold determines a singular foliation of
the Euclidean or Lorentzian space, where each leaf is either an isoparametric subma-
nifold or a focal manifold of an isoparametric submanifold.
The rank of an iso parametric submanifold of a Euclidean space is the maximal
number of linearly independent curvature normals. If the isoparametric submanifold
M is full, then its rank coincides with the codimension in the Euclidean space. In
fact, if the codimension of M would be bigger, then there would exist a nontrivial
parallel normal vector field
η
on M which is perpendicular to any curvature normal
and so A
η
= 0. This implies that we can reduce the codimension of M.
4.2.2 The Coxeter group
We will now associate a Coxeter group with any complete isoparametric subma-
nifold of a Euclidean space (cf. [275]). A similar construction can be carried out in
the Lorentzian situation.
Let M be a complete isoparametric submanifold of R
n
.Forp, q ∈ M we denote
by
τ
p,q
:
ν
p
M →
ν
q
M the ∇
⊥
-parallel transport. The affine parallel transport
˜
τ
p,q
: p +
ν
p
M → q +
ν
q
M
is the unique isometry defined by
˜
τ
p,q
(p)=q and d
p
˜
τ
p,q
=
τ
p,q
.SinceM is isopara-
metric, we have
˜
τ
p,q
(F
M
(p)) = F
M
(q),
where F
M
(p) is the focal set of M at p.Let
σ
p
i
be the orthogonal reflection of the
affine space p +
ν
p
M in the hyperplane
i
(p), i ∈{1,...,g}.
Lemma 4.2.6 The point
σ
p
i
(p) is the antipodal point of p in the sphere S
i
(p) and
˜
τ
p,
σ
p
i
(p)
=
σ
p
i
.
Proof Since the curvature normal n
i
(p) is in the radial direction of S
i
(p),thefirst
part is easily verified. Let us compute
τ
p,
σ
p
i
(p)
. Consider a curve c(t) in S
i
(p) from
p to
σ
p
i
(p). Recall that S
i
(p) is totally ge odesic in M and invariant under the shape
Isoparametric Submanifolds and Their Focal Manifolds 143
operator of M (Lemma 3.4.2). Therefore, parallel transport in the normal space of
S
i
(p) (regarded as a submanifold of R
n
) restricted to vectors in
ν
p
M coincides with
parallel transport in the normal space of M. Since parallel transport in the sphere
maps n
i
(p) to n
i
(
σ
p
i
(p)) = −n
i
(p),wehave
τ
p,
σ
p
i
(p)
(n
i
(p)) = −n
i
(p). Moreover, if
ξ
is in the orthogonal complement of n
i
(p) in
ν
p
M and
ξ
(t) its parallel transport
along c(t) with respect to the normal connection, we have
d
dt
ξ
(t)=−A
S
i
(p)
ξ
(t )
˙c(t)+∇
⊥
˙c(t)
ξ
(t)=0,
so
ξ
(t) is constant and
τ
p,
σ
p
i
(p)
(
ξ
)=
ξ
. Thus, d
p
˜
τ
p,
σ
p
i
(p)
=
τ
p,
σ
p
i
(p)
= d
p
σ
p
i
.
We now relate antipodal maps with the focal structure. For every i ∈{1,...,g}
we renormalize the curvature normal n
i
to
ψ
i
=
2
n
i
,n
i
n
i
.
Since n
i
is a parallel normal vector field on M,
ψ
i
is also a parallel normal vector
field on M.Then
ϕ
i
: M → M , p → p +
ψ
i
(p)=
σ
p
i
(p)
is a diffeomorphism of M mapping a point p to its antipodal point
σ
p
i
(p) on the
curvature sphere S
i
(p). For this reason the map
ϕ
i
is also called the antipodal map
or involution on M associated with the curvature distribution E
i
.SinceM is isopara-
metric, we have
˜
τ
p,
ϕ
i
(p)
(F
M
(p)) = F
M
(
ϕ
i
(p)). Note that
ϕ
i
can also be regarded as
the projection map sending M to the par allel manifold M
ψ
i
(which coincides with
M). From Proposition 3.4.11 we get F
M
(p)=F
M
ψ
i
(p +
ψ
i
(p)).SinceM = M
ψ
i
and
ϕ
i
(p)=p +
ψ
i
(p), we conclude that
˜
τ
p,
ϕ
i
(p)
(F
M
(p)) = F
M
(p). Therefore, the reflec-
tions
σ
p
i
permute the focal hyperplanes
1
(p),...,
g
(p) and generate a finite group
of reflections [156].
Theorem 4.2.7 (Terng) Let M be a complete isoparametric submanifold of R
n
and
p ∈M. The orthogonal reflections
σ
p
i
of the affine normal space p+
ν
p
Minthefocal
hyperplanes
i
(p) generate a finite reflection group W
p
⊂ O(p +
ν
p
M).
The finite reflection group W
p
in Theorem 4.2.7 is called the Coxeter group of
Matp. Note that two Coxeter groups W
p
and W
q
are conjugate because parallel
transport along any curve from p to q conjugates W
p
to W
q
. We then write W for W
p
and call W the Coxeter group of M.
Note that the reflections
σ
p
i
determine perm utations of the curvature distribu-
tions. To see this, let c(t) be a (piecewise differentiable) curve in M based at p and
˜c(t)=c(t)+
ψ
i
(t) a curve based at
ϕ
i
(p).Then
ν
p
M =
ν
ϕ
i
(p)
M, and by Lemma 3.4.6
a normal vector field
ζ
along c(t) is ∇
⊥
-parallel if and only if it is ∇
⊥
-parallel along
˜c(t). Moreover
A
ζ
ϕ
i
(p)
= A
ζ
p
◦(id −A
ψ
i
(p)
)
−1
= A
ζ
p
◦d
p
ϕ
−1
i
.
144 Submanifolds and Holonomy
It follows that there there exists a permutation
σ
i
of {1, ..., g} such that
d
p
ϕ
i
(E
j
(p)) = E
σ
i
( j)
(
ϕ
i
(p)). (4.3)
Note that E
j
(p) equals E
σ
i
( j)
(
ϕ
i
(p)) as linear subspace of R
n
. The curvature normals
at
ϕ
i
(p) are given by
n
σ
i
( j)
(
ϕ
i
(p)) =
1
1 −
ψ
i
(p),n
j
(p)
n
j
(
ϕ
i
(p)). (4.4)
Remark 4.2.8 The above relations im pose severe restrictions on the geometry of M.
Indeed, as a consequence of (4.3), we get m
i
= rk(E
i
)=rk(E
σ
i
( j)
)=m
σ
i
( j)
.
In general, when G is a reflection group that is generated by reflections in hyper-
planes
π
j
orthogonal to vectors
ν
j
, one defines the rank of G as the maximal number
of linearly independent vectors in {
ν
j
}. Thus, the rank of the Coxeter group W asso-
ciated with a complete isoparametric submanifold M of R
n
is equal to the rank of the
isoparametric submanifold M.IfM is a homogeneous isoparametric submanifold,
we will see in Section 4.4 that M is a principal orbit of the isotropy representation o f
a Riemannian symmetric space. The Coxeter group of the isoparametric submanifold
then coincides with the Weyl group of the symmetric space.
The complement in p +
ν
p
M of the union of the focal hyperplanes
i
(p) is not
connected. Let C be one of its connected components. Its closure
C is a simplicial
cone and a fundamental domain for the W -action on p +
ν
p
M, that is, each W -orbit
meets
C at exactly one point. The closure C is called a Weyl chamber for W .
Example 4.2.1 (Isoparametric hypersurfaces in spheres) Let M be a compact
isoparametric hypersurface in the sphere S
n
.ThenM has codimension 2 in R
n+1
and its associated Coxeter g roup W has rank 2 and therefore is a reflection g roup of
the plane. By the classification of finite Coxeter groups (see, e.g., [156, Chapter 2])
or by direct inspection, W is the dihedral group of order 2g (group of symmetries of
a regular g-gon), where g is the number of different curvature normals. The picture
below illustrates the case g = 3:
T
T
T
T
T
T
T
T
1
2
3
If g is odd, the Coxeter group is transitive on the set of focal lines and hence the
multiplicities are all equal. If g is even, there are two orbits and two multiplicities. If
we arrange indices so that
i
and
i+1
are adjacent, then m
i
= m
i+2
for all i modulo g
( [222], see also Section 2.9).
By the remarkable work of M¨unzner in [222, 223], which uses delicate cohomo-
logical arguments, the number g of curvature normals can only be 1, 2, 3, 4 or 6. In
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