128 Submanifolds and Holonomy
3. Focal manifolds.LetM
ξ
be a parallel focal manifold with respect to a parallel
normal isoparametric section
ξ
on a submanifold M of R
n
and
π
: M → M
ξ
the
canonical projection. The fiber
π
−1
({q}), q ∈ M
ξ
, is contained in the affine normal
space q +
ν
q
M
ξ
and, by Lemma 3.4.10, is invariant under ∇
⊥
-parallel transport in
ν
M
ξ
. By Lemma 3.4.10(c) we see that M can be regarded locally as a partial tube
about M
ξ
. This actually generalizes the situ ation of an isoparametric submanif old
and one of its focal manifolds.
4. Holonomy tubes.Let(M)
η
p
be a holonomy tube around a submanifold M .If
η
p
< fd(M),then(M)
η
p
is a p artial tube and the typical fiber is the orbit of
η
p
under the action of the normal holonomy group.
Let B be a partial tube about M and let
˜
γ
beacurveinB joining ˜p =(p,
ϕ
p
(x)) and
˜q =(q,
ϕ
q
(y)). We project
˜
γ
down to M and get a curve
γ
from p and q. Writing
˜
γ
(t)=
γ
(t),
ϕ
γ
(t )
(
σ
(t))
determines a curve
σ
(t) in the fiber S. Again, parallel transport
along
σ
composed with a fixed isometry
θ
x
: R
d
→
ν
x
S determines an isometry
θ
y
:
R
d
→
ν
y
S,whered is the codimension of S in R
k
. The following result ( [65], Lemma
4.2) relates parallel transport along
˜
γ
in the normal bundle to a partial tube to ∇
⊥
-
parallel transport in M along
γ
and the parallel transport in the normal bundle of the
typical fiber along
σ
:
Proposition 3.4.18 Parallel transport in the normal bundle of the partial tube B
along
˜
γ
is given by
ϕ
p
θ
x
w →
ϕ
q
θ
y
w.
The proof can be found on page 158 in [65] and uses arguments similar to that
used in the proof of Lemma 3.4.6 together with the description of the normal space
to the partial tube we gave above. We propose it as an exercise (Exercise 3.6.3).
Note that this proposition generalizes Lemma 3.4.6. Indeed, if M
ξ
is a parallel
focal manifold of M with respect to a parallel normal isoparametric section
ξ
,as
in Example 3 above, we can write a point of the partial tube M as (p,
ξ
(p)),and
the typical fiber can be identified with the set of vectors
ξ
(q) such that q +
ξ
(q)=
p +
ξ
(p).Since
ξ
is ∇
⊥
-parallel, we have
ϕ
q
ξ
(p)=
ξ
(q), so that the curve
σ
can be
identified with the constant curve t →
ξ
(p).
3.5 Further remarks
3.5.1 Realizations of s-representations as normal holonomy groups
Heintze and Olmos computed in [146] the normal holonomy of orbits of s-
representations and showed that all s-representations arise as normal holonomy rep-
resentations with 11 exceptions. Up to now, no example has been found of a sub-
manifold realizing one of these exceptions as normal holonomy representation. The
The Normal Holonomy Theorem 129
simplest exception, which has rank one, is the isotropy representation of the Cay-
ley projective plane OP
2
= F
4
/Spin
9
. Tezlaff [318] gave a negative answer to the
question whether this representation is the normal holonomy representation of one
of the focal manifolds of the inhomogeneous isoparametric hypersurfaces in spheres
of Ferus, Karcher and M¨unzner [131], which would have been good candidates.
A still open conjecture states that if M is a full irreducible homogeneous sub-
manifold of the sphere that is not the orbit of an s-representation, then the normal
holonomy group acts transitively on the unit sphere of the normal space (see Conjec-
ture 5.2.14).
3.5.2 Normal holonomy and irreducibility
The Normal Holonomy Theorem 3.2.1 provides, in particular, an orthogonal de-
composition of the n ormal space at p into invariant subspaces. However, the exis-
tence of an invariant subspace for the normal holonomy does not imply in g eneral
that the submanifold splits locally (both extrinsically and intrinsically). For exam-
ple, for a submanifold of Euclidean space contained in a sphere, the line determined
by the position vector p is an invariant subspace under normal holonomy (it always
belongs to the flat part of
ν
M), but such a submanifold does not necessarily split.
For submanifolds of space forms one can get only weaker versions of de Rham’s
Decomposition Theorem (see, for instance, Exercise 3.6.1). Di Scala proved in [104]
for complex submanifolds M of C
n
a versio n of the de Rham Decomposition The-
orem: if Φ splits, then M locally splits as a product of submanifolds (see Section
7.4).
3.5.3 A bound on the number of factors of the normal holonomy
representation
Let M be a submanifold of R
n
(or, more generally, of a space of constant curva-
ture). Let p ∈ M and let
ν
p
M = V
0
⊕V
1
⊕...⊕V
k
be the decomposition of
ν
p
M given by the Normal Holonomy Theorem (applied to
the local normal holonomy group). The following result gives a sharp bound for the
number of irreducible factors of the normal holonomy representation.
Theorem 3.5.1 (Olmos, Ria
˜
no-Ria
˜
no [263]) Let M be an m-dimensional submani-
fold of R
n
(or, more generally, of a space of constant curvature). Assume that at any
point of M the local and the restricted normal holonomy groups coincide (or, equiv-
alently, the dimensions of the local normal holonomy groups are constant on M ). Let
p ∈ M and k be the number of irreducible (nonabelian) subspaces of the representa-
tion of the restricted normal holonomy group Φ
p
on
ν
p
M. Then k ≤ [
m
2
]. Moreover,
this bound is sharp for all m ∈ N (also in the class of irreducible submanifolds).
130 Submanifolds and Holonomy
Proof Let
ν
p
M = V
0
⊕V
1
⊕...⊕V
k
be the d ecomposition given by the Normal Holonomy Theorem. The local holon-
omy group Φ
p
acts trivially on V
0
and irreducibly on V
i
for i ∈{1,...,k}.Fromthe
assumptions we obtain that V
i
extends to a ∇
⊥
-parallel subbundle
ν
i
of the normal
bundle
ν
M, i = 0,...,k (by possibly making M smaller around p). Note that we
have the decomposition
ν
M =
ν
0
⊕
ν
1
⊕...⊕
ν
k
. Moreover, from the assumptions
it follows that Φ
q
acts trivially on
ν
0
q
and irreducibly on
ν
i
q
for all i ∈{1,...,k} and
q ∈ M.
Let R
⊥
(
ξ
,
ξ
) be the adapted normal curvature tensor. From the expression of
R
⊥
in terms of the shape operator A we get R
⊥
(
ξ
,
ξ
)=0 if and only if [A
ξ
,A
ξ
]=0.
Note that for i = j we have R
⊥
(
ξ
i
,
ξ
j
)=0if
ξ
i
,
ξ
j
are normal sections that lie in
ν
i
and
ν
j
, respectively.
There must exist a point q ∈ M, arbitrary close to p, such that R
⊥
(
ν
i
q
,
ν
i
q
) =
{0} for all i ∈{1,...,k}. In fact, there exists q
1
∈ M, arbitrary close to p, such that
R
⊥
(
ν
1
q
1
,
ν
1
q
1
) = {0} (otherwise,
ν
1
would be flat). The above inequality must be true
in a neighborhood Ω
1
of q
1
. Now choose q
2
∈ Ω
1
such that R
⊥
(
ν
2
q
2
,
ν
2
q
2
) = {0}.
Continuing with this process we find q = q
k
with the desired properties.
Let us show that for any i ∈{1,....,k} there exist
ξ
i
,
ξ
i
∈
ν
i
q
such that [A
ξ
i
,A
ξ
i
]
does not belong to the algebra of endomorphisms generated by {A
η
i
},where
η
i
∈
ν
q
has no component in
ν
i
q
. In fact, if this is not true, then [A
ξ
i
,A
ξ
i
] commutes with A
ξ
i
for any
ξ
i
,
ξ
i
in
ν
i
q
(M) (since the shape operators of elements of the subspaces
ν
j
q
commute with A
ξ
i
,if j = i). Then
[[A
ξ
i
,A
ξ
i
],A
ξ
i
],A
ξ
i
= 0 = −[A
ξ
i
,A
ξ
i
],[A
ξ
i
,A
ξ
i
]
and hence [A
ξ
i
,A
ξ
i
]=0. This is a contr adiction since R
⊥
(
ν
i
q
,
ν
i
q
) = {0}, and hence
our assertion is proved.
Note that [A
ξ
1
,A
ξ
1
],...,[A
ξ
k
,A
ξ
k
] are linearly independent and commuting
skewsymmetric endomorphisms o f T
q
M.Thenk ≤ rk(SO(T
p
M)) = [
m
2
] (the integer
part of
m
2
). This proves the in equality.
Let us show that the inequality is sharp. Let M
2
beasurfaceinS
k
1
−1
and
¯
M
3
be a
3-dimensional submanifold of S
k
2
−1
such that their normal holonomy representations
have one irreducible factor (for example, the Veronese submanifolds V
2
and V
3
,
cf. [263]). Let m > 3 and write m = 2d if m is even or m = 2d + 3ifm is odd. Let M
be equal to the d-fold Riemannian product M
d
of M
2
or to M
d
×
¯
M
3
. By construction,
M is contained in the product of Euclidean spaces. The number o f irreducible factors
of the normal holonomy group (representation) of M is exactly the upper bound [
m
2
].
Since M is contained in a sphere, we can apply to M a conformal transformation of
the sphere (the normal holonomy group is a conformal invariant) in such a way that
M is an irreducible (Riemannian) submanifold o f the Euclidean space.
The Normal Holonomy Theorem 131
3.5.4 Normal holonomy of surfaces
Using the bound of Theorem 3.5.1 on the number of normal holonomy repre-
sentation components and properties of holonomy systems we will now prove the
following result.
Theorem 3.5.2 Let M be a surface in a Euclidean space with the property that
around any point it is not contained in a sphere or in a proper affine subspace. Then
the local normal holonomy group is either trivial or it acts transitively on the unit
sphere of the normal space.
Proof Assume that the local normal holonomy g roup Φ
loc
p
is not trivial and let p ∈
M. We will show that Φ
loc
p
acts transitively on the unit sphere of the normal space
ν
p
M.
Note first that there are no parallel nontrivial umbilical normal sections, o r else M
would be contained either in a sphere or in an affine subspace. Furthermore, the factor
V
0
(that is, the fixed point set of the normal holonomy group) is trivial, for otherwise
there should exist a nonumbilical parallel normal vector field
ξ
(around arbitrary
close points to p). This is impossible, because it would imply that the normal bundle
is flat. In fact, A
ξ
would commute with all other shape operators and by the Ricci
equations all shape operators would commute, since dimM = 2, and so R
⊥
= 0.
The bound given in Theorem 3.5.1 forces the local normal holonomy group to act
irreducibly. We now claim that Φ
loc
p
is transitive on the unit sphere of
ν
p
M. Suppose
that this is not the case. Then there exists a point q arbitrarily close to p such that
R
⊥
q
= 0andso[
ν
q
M,R
⊥
q
,Φ
loc
q
] is an irreducible nontransitive holonomy system.
This holonomy system is symmetric by Theorem 3.3.7. In particular, the first normal
space N
1
q
coincides with
ν
q
M. This is because, otherwise, there would exist
ξ
∈
ν
q
M
with A
ξ
= 0andsoR
⊥
q
(
ξ
,·)=0, contradicting irreducibility. The map
ξ
→ A
ξ
is
injective and so we have that dim
ν
q
M ≤ 3 (note that the dimension of the space
of 2 ×2 symmetric matrices is 3). Now, it is not difficult to see that an irreducible
symmetric space of dimension at most 3 must be of rank one (Exercise 3.6.18, or
else use the classification of symmetric spaces, cf. Appendix). This means the normal
holonomy group is transitive on the unit sphere of the normal space.
Note that, for a surface contained in a sphere but not contained in a proper affine
subspace (or, equivalently in a smaller dimensional sphere), there is an analogous
result (Exercise 3.6.16).
3.5.5 Computing the normal holonomy group
The description of the Lie algebr a L (Φ
p
) given by the Ambrose-Singer Theo-
rem is not very explicit since the normal holonomy algebra depends also on parallel
transport
τ
⊥
γ
. Thus it is not very useful for explicit computations. In some cases, like
homogeneous submanifolds, one can compute the normal holonomy group by taking
the covariant derivatives of the normal curvature tensor.
132 Submanifolds and Holonomy
Proposition 3.5.3 Let M be a homogeneous submanifold of a space form. The Lie
algebra L (Φ
p
) of the normal holonomy group is generated by the skewsymmetric
operators on
ν
p
Moftheform
(∇
⊥
R
⊥
)
k
V
1
...V
k
(X,Y ),
where X ,Y,V
1
,...,V
k
∈ T
p
M and k ≥ 0.
Proof See [178, vol. I, Theorem 9.2, page 152] for a proof in the general case of a
linear connection.
Example 3.5.1 (Normal holonomy of SO
n
⊂ gl
n
(R)
∼
=
R
n
2
.) Using the above propo-
sition one can see that
L (Φ
∗
I
)=span {R
⊥
(A
∗
,B
∗
) : A, B ∈ so
n
} = so
n
(Exercise 3.6.19, cf. [47]). Note that SO
n
⊂ gl
n
(R)
∼
=
R
n
2
is an orbit of an s-
representation (namely, of the symmetric space SO
2n
/SO
n
SO
n
).
In Section 5.2 we will give a description of normal holonomy of orbits in terms
of projection of Killing vector fields (Theorem 5.2.7). This yields a very practical
tool for computing the normal holonomy group of a homogeneous submanifold.
3.6 Exercises
Exercise 3.6.1 This exercise is a sort of extrinsic version of de Rham’s Decomposi-
tion Theorem. Let M be a submanifold of a space form
¯
M
n
(
κ
) and Φ
c
be the com-
bined holonomy (see page 95). Let p ∈ M and suppose that both T
p
M and
ν
p
M split
as orthogonal direct sums T
p
M = T
1
⊕T
2
and
ν
p
M =
ν
1
⊕
ν
2
and that Φ
c
splits as a
product Φ
1
×Φ
2
with Φ
1
acting trivially on T
2
⊕
ν
2
and Φ
2
acting trivially on T
1
⊕
ν
1
.
Assume further that the second fundamental form
α
of M satisfies
α
(T
1
,T
1
) ⊂
ν
1
and
α
(T
2
,T
2
) ⊂
ν
2
. Prove that M is locally reducible.
Exercise 3.6.2 We use the same notation as on page 102 and define the loop
γ
s,t
by
the following procedure: Move on the coordinate lines from f (0, 0) to f (s,0),then
to f (s,t), then back to f (0,t) and finally back to f (0, 0). Prove that
τ
⊥
γ
s,0
ξ
=
τ
⊥
γ
0,t
ξ
=
ξ
and that
R
⊥
(u,v)
ξ
= −
∂
2
∂
s
∂
t
(s,t)=(0,0)
τ
⊥
γ
s,t
ξ
.
Conclude that
R
⊥
(u,v)
ξ
= −
1
2
d
2
dt
2
t=0
τ
⊥
γ
t,t
ξ
.
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