The Normal Holonomy Theorem 113
3.4 Some geometric applications of the Normal Holonomy
Theorem
3.4.1 Parallel normal isoparametric sections
Let M be a submanifold of a standard space form
¯
M
n
(
κ
).
Definition 3.4.1 A p arallel normal vector field
ξ
on M is called a parallel normal
isoparametric section if the shape operator A
ξ
has constant eigenvalues.
We will m a inly focus here on submanifolds of R
n
, a situation that includes sub-
manifolds of spheres by regarding the latter as submanifolds of R
n
. Most of the
results extend to submanifolds of the real hyperbolic space, the latter viewed as a
Riemannian hypersurface of the Lorentzian space (cf. [266]).
The constancy of the eigenvalues of the shape operator A
ξ
is a tensorial property,
for it is equivalent to the higher order mean curvatures in direction
ξ
being constant
(see Remark 3.1.2). Recall also that, if M is full in R
n
, by Theorem 1.5.1 on the
reductio n of codimension, then the only parallel umbilical isoparametric section s are
constant multiples of the position vector field with respect to a suitable origin. This
situation is of course trivial. Thus we will be interested in nonumbilical isoparametric
sections and we will actually see that the existence of one such section has strong
consequences on the geometry of the submanifold. A first example in this direction
is the following useful lemma.
Let
ξ
be a parallel normal isoparametric section,
λ
1
,...,
λ
g
be the different con-
stant eigenvalues of A
ξ
and TM = E
1
⊕...⊕E
g
be the decomposition into the smooth
curvature distributions of the shape operator A
ξ
.
Lemma 3.4.2 Let M be a submanifold of R
n
and
ξ
a be parallel normal isopara-
metric section.
(a) Each curvature distribution E
i
of A
ξ
is autoparallel and hence integrable with
totally geodesic leaves.
(b) Each curvature distribution E
i
is invariant under all shape operators of M.
(c) If S
i
(p) is the leaf of E
i
containing p ∈ M, then S
i
(p) is contained in the affine
subspace p +(E
i
(p) ⊕
ν
p
M) of R
n
and A
i
ξ
= A
ξ
E
i
(p)
for all
ξ
∈
ν
p
M, where
A
i
is the shape operator of S
i
(p) in p +(E
i
(p) ⊕
ν
p
M).
Proof (a): The statement is trivial if g = 1, so we assume that g ≥ 2. Let X ,Y be
sections in E
i
and Z be a section in E
j
with i = j. Using the Codazzi equation we get
(
λ
i
−
λ
j
)∇
X
Y,Z = (∇
X
A
ξ
)Y,Z = (∇
Z
A
ξ
)X,Y = 0.
Since this holds for all i = j this implies that ∇
X
Y is a section in E
i
, which means
that E
i
is an autoparallel subbundle of TM.
114 Submanifolds and Holonomy
(b): Since ∇
⊥
ξ
= 0, we have R
⊥
(X,Y )
ξ
= 0 and the Ricci equatio n implies that
[A
ξ
,A
η
]=0 for all normal vector fields
η
. Thus, all shape operators commute with
A
ξ
and consequently preserve the curvature distributions.
(c) The statement is trivial if g = 1, so we assume that g ≥2. Let c : [0, 1] →S
i
(p)
be a smooth curve with c(0)=p, v be a normal vector of S
i
(p) ⊂ M at p and V be
the parallel normal vector field with initial value V (0)=v along c with respect to the
normal connection of S
i
(p) ⊂M.Since
α
( ˙c(t),V (t)) = 0andS
i
(p) is totally geodesic
in M, the covariant derivative of V with respect to the Levi-Civita connection on R
n
vanishes. This implies that S
i
(p) is contained in the affine space p +(E
i
(p) ⊕
ν
p
M).
The statement about the shape operator A
i
follows from the fact that S
i
(p) is totally
geodesic in M.
3.4.2 Parallel manifolds and focal manifolds
Let M be an m-dimensional submanifold of R
n
and assume that there exists a
nontrivial parallel normal vector field
ξ
on M. We will consider
ξ
also as a smooth
map M → R
n
.Wedefine a smooth map
t
ξ
: M → R
n
, p → p +
ξ
(p).
The differential of t
ξ
at p has the same rank as id
ν
p
M
−A
ξ
(p)
. We denote by
ν
(p)
the dimension of the kernel of id
ν
p
M
−A
ξ
(p)
.Thenp +
ξ
(p) is a focal point of M in
direction
ξ
if
ν
(p) > 0, and then
ν
(p) is the multiplicity of the focal p oint.If
ν
(p)=
ν
is constant, which happens for instance if
ξ
is a parallel normal isoparametric
section, then the image
M
ξ
= t
ξ
(M)={p +
ξ
(p) : p ∈ M}
of t
ξ
is an immersed submanifold of R
n
with dimension dim M −
ν
(cf. [70, 282]). If
ν
(p)=
ν
> 0, then M
ξ
is called a focal (or parallel focal) manifold of M in direc-
tion
ξ
.If
ν
= 0, or equivalently, if t
ξ
is a regular map, then M
ξ
is called a parallel
manifold of M in direction
ξ
. We illustrate these concepts in Figures 3.2 and 3.3.
In this section we study lo cal properties and hence we can assume that M
ξ
is em-
bedded. Global properties will be studied in Section 4.5 (see in particular Exercises
4.6.6, 4.6.7 and 4.6.8) and in Chapter 5.
Remark 3.4.3 If M is a submanifold of S
n
, we can consider M as a submanifold of
R
n+1
andthendefine parallel and focal manifolds of M via this approach. Note that
every parallel normal vector field of M ⊂ S
n
is also a parallel normal vector field of
M ⊂ R
n+1
. One can easily verify that parallel and focal manifolds of submanifolds
of S
n
are also contained in a sphere (of different radius in general).
Remark 3.4.4 We can also replace the Euclidean space by a Lorentzian space and
consider the hyperbolic space H
n
as a Riemannian hypersurface in R
n,1
in line with
our standard model. Also in this situation every parallel normal vector field of M ⊂
H
n
is a parallel normal vector field of M ⊂R
n,1
.ViewingM as a submanifold of R
n,1
The Normal Holonomy Theorem 115
FIGURE 3.2: A piece of a cylinder M with its parallel displacement in direction of
vector fields
ξ
and
ξ
+
ζ
pointing inward. M
ξ
a parallel manif old, while M
ξ
+
ζ
is
focal.
FIGURE 3. 3: A piece of a cyclides of Dupin and two of its parallel surfaces.
has the advantage that we can still work in a vector space. Thus p +
ξ
(p) is a focal
point of M ⊂ R
n,1
if ker(id
ν
p
M
−A
ξ
(p)
) = {0}.
Let M
ξ
be an embedded parallel or focal manifold of M. Then the smooth map
π
: M → M
ξ
, p → t
ξ
(p)=p +
ξ
(p)
is a submersion, and a diffeomorphism if M
ξ
is a parallel manifold. We sometim es
write
π
ξ
: M → M
ξ
for this submersion to indicate the dependence o n
ξ
. We denote
by H and V the horizontal and vertical distributions induced by
π
. Note that H
p
is
isomorphic and parallel to T
π
(p)
M
ξ
and
ν
p
M ⊂
ν
π
(p)
M
ξ
. Moreover , we have
V
p
= ker(d
p
π
)=ker(id −A
ξ
(p)
)=T
p
π
−1
({
π
(p)}),
that is, the vertical distribution is formed by the tangent spaces of the fibers of the
map
π
: M → M
ξ
. The horizontal distribution is just the orthogonal complement in
TM of the vertical distribution. We illu strate this in Figure 3.4.
116 Submanifolds and Holonomy
FIGURE 3. 4: Submanifold, focal manifold, and horizontal direction.
Example 3.4.1 (Focalization of a curvature distribution) Let M be a submanifold
of R
n
and
ξ
be a parallel normal isoparametric section on M.Let
λ
i
be a nonzero
eigenvalue of A
ξ
.Then
ξ
i
=
λ
−1
i
ξ
is a parallel normal isoparametric section with
ker(id −A
ξ
i
)=E
i
. Denote by
π
i
: M →M
ξ
i
the focal map from M onto M
ξ
i
. Then the
tangent space of M
ξ
i
at
π
i
(p) is given by
T
π
i
(p)
M
ξ
= E
1
(p) ⊕...⊕
ˆ
E
i
(p) ⊕...⊕E
g
(p),
where
ˆ
E
i
(p) means that E
i
(p) is o mitted. So the vertical and horizo ntal d istribution
with respect to
π
i
: M → M
ξ
i
is given by V
i
= E
i
and H
i
= E
1
⊕...⊕
ˆ
E
i
⊕...⊕E
g
,
respectively. We will say that M
ξ
i
is the focal manifold that focalizes E
i
(or
λ
i
).
The next lemma describes a fundamental property of a focal map
π
: M → M
ξ
.
Lemma 3.4.5 The fibers of the projection
π
from M onto a parallel focal manifold
M
ξ
are totally geodesic submanifolds of M and the shape operator of M leaves the
orthogonal decomposition T M = H ⊕V invariant.
Proof The statement is trivial if M
ξ
is a point, so we can assume that dimM
ξ
≥ 1.
Let X ,Y be sections in V and Z be an eigenvector of A
ξ
with A
ξ
Z =
λ
Z,
λ
= 1.
Using the Codazzi equation we get
(1 −
λ
)∇
X
Y,Z = (∇
X
A
ξ
)Y,Z = (∇
Z
A
ξ
)X,Y = 0.
Since H is spanned by such eigenvectors Z,thisimpliesthat∇
X
Y is a section in
V , which means that V is an autoparallel subbundle of TM and hence its leaves are
totally geodesic submanifolds of M. The proof for the second statement is analogous
to the one for Lemma 3.4.2 (b).
The Normal Holonomy Theorem 117
Let c be a curve in M
ξ
and q = c(t
o
). Then there exists for each point p ∈
π
−1
({q}) exactly one curve ˜c in M with ˜c(t
o
)=p,
π
◦ ˜c = c and
˙
˜c(t) ∈ H
˜c(t)
for
all t (sufficiently close to t
o
). This curve ˜c is called the horizontal lift of c through p.
By definition of
π
,wehave
˜c(t)+
ξ
( ˜c(t)) = c(t).
Differentiating this equation gives
(id −A
ξ
( ˜c(t))
)
˙
˜c(t)=
˙
˜c(t) −A
ξ
( ˜c(t))
˙
˜c(t)= ˙c(t). (3.6)
The next result is of g reat importance when comparing the geometry of a sub-
manifold M with the one of a parallel (possibly focal) manifold M
ξ
(see [147, page
170]).
Lemma 3.4.6 Let c be a curve in M
ξ
and q = c(t
o
).Let ˜c be the horizontal lift of c
through p ∈
π
−1
({q}).Foreach
η
∈
ν
p
M ⊂
ν
q
M
ξ
the parallel transports of
η
along
c and ˜c with respect to the normal connections of M
ξ
and M, respectively, coincide.
Proof The parallel transport
η
(t) of
η
along ˜c(t) with resp ect to the normal co nnec-
tion of M is orthogonal to M
ξ
since
ν
M ⊂
ν
M
ξ
. Since the horizontal distribution is
invariant under the shape operator A of M by Lemma 3.4.5, we get
η
(t)=−A
η
(t )
˙
˜c(t) ∈ H
˜c(t)
∼
=
T
c(t)
M
ξ
.
Thus
η
(t) is a parallel normal vector field on M
ξ
along c(t). Conversely, it easy to
see that if
η
(t) is parallel along c(t) with respect to th e normal connection of M
ξ
,
then it is parallel along ˜c(t) with respect to the normal connection of M.
We now compare the shape operators A and
ˆ
A of M and M
ξ
respectively. We have
the following important relation between the shape operators in the common normal
directions to M and M
ξ
.
Lemma 3.4.7 (Tube formula) For a l l
η
∈
ν
p
M ⊂
ν
π
(p)
M
ξ
we have
ˆ
A
η
=(A
η
H
) ◦
(id −A
ξ
(p)
)
H
−1
.
Proof Let c beacurveinM
ξ
with ˙c(0) ∈ T
π
(p)
M
ξ
∼
=
H
p
.Let ˜c be the horizontal lift
of c through p and set X =
˙
˜c(0). Denote by
η
(t) the p arallel transport of
η
along ˜c
with respect to the normal con nection of M. By Lemma 3.4.6,
η
(t) is also the parallel
transport of
η
along c with respect to the normal connection of M
ξ
.Using(3.6)we
get
A
η
X = A
η
˙
˜c(0)=−
η
(0)=
ˆ
A
η
˙c(0)=
ˆ
A
η
(id −A
ξ
(p)
)X,
which implies the assertion.
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