Basics of Submanifold Theory in Space Forms 11
The second model is known as the Poincar
´
e ball model and is given by the open
ball
{x ∈ R
n
: x < 1}
with the Riemannian metric
ds
2
= 4
dx
2
1
+ ...+ dx
2
n
1 −dx
2
1
−...−dx
2
n
.
In this model, the geodesics are circles that are orthogonal to the boundary sphere
{x ∈ R
n
: x = 1} of the ball (including the degenerate circles given by diameters).
1.2.4 The classification problem for space forms
Let n ≥ 2. The Riemannian manifold
¯
M
n
(
κ
)=
⎧
⎪
⎨
⎪
⎩
S
n
(
κ
−1/2
) if
κ
> 0,
R
n
if
κ
= 0,
H
n
((−
κ
)
−1/2
) if
κ
< 0,
is connected and simply connected and often referred to as a standard space of con-
stant curvature
κ
. A connected Riemannian manifold M
n
of constant curvature
κ
is
called a space form, or sometimes also real space form to distinguish it from complex
and quaternionic space forms. It is called a spherical space form,aflat space form,
or a hyperbolic space form depending on whether
κ
> 0,
κ
= 0, or
κ
< 0, respec-
tively. Any space form M
n
of constant curvature
κ
admits a Riemannian covering
map
¯
M
n
(
κ
) → M
n
. A classical problem is to determine all compact space forms. A
theorem by Bieberbach says that any compact flat space form M is covered by a flat
torus, where the group of deck transformations is a free abelian normal subgroup of
the first fundamental group
π
1
(M) of M with finite rank. The spherical space forms
were classified by Wolf [347]. The even-dimensional case appears to be quite simple,
as one can show that any even-dimensional spherical space form is isometric either to
the sphere or to the real projective space of corresponding dimension and curvature.
The theory of hyperbolic space forms is more subtle and still an active research field.
1.3 Principal curvatures
The shape operator or second fundamental form is the fundamental object in
submanifold theory. Practically all geometrical problems concerning submanifolds
involve them in one or another way. In the course of this book, we will deal with
submanifolds whose second fundamental form has a “regular” behavior especially to
what concerns its eigenvalues, called principal curvatures.
12 Submanifolds and Holonomy
Various properties of
α
or A lead to interesting classes of submanifolds. For in-
stance, the vanishing of
α
leads to totally geodesic submanifolds, which will be
discussed later in Section 1.4.
The mean curvature vector field H of an m-dimensional submanifold M of
¯
M is
defined by
H =
1
m
tr(
α
),
and h = H is the mean curvature function of M.Aminimal submanifold is a sub-
manifold with vanishing mean curvature function. This class o f submanifolds has
already attracted mathematicians for a lon g time. There is a great variety of literature
concerning minimal submanifolds and, in particular, minimal surfaces. We refer the
interested reader to [103].
A simple condition for principal curvatures on a hypersurface M is that they sat-
isfy some functional relation, in which case M is called a Weingarten hypersurface.
This is a classical topic and for a modern treatment of the subject we refer to [275].
For higher codimension, Terng generalized this notion by requiring that the subma-
nifold has a flat normal bundle and the principal curvatures satisfy a polynomial
relation [317] (see Exercise 1.8.5).
In the course of this book we will encounter various kinds of properties of the
second fundamental form or the shape operator that lead to interesting areas of math-
ematics. We start with discussing principal curvatures in more detail.
1.3.1 Principal curvatures and principal curvature vectors
Let M be a submanifold of a space form
¯
M. As usual, the shape operator of M is
denoted by A and the second fundamental form by
α
. Recall that A and
α
are related
by the equation
α
(X,Y ),
ξ
= A
ξ
X,Y ,
where X,Y ∈ T
p
M and
ξ
∈
ν
p
M, p ∈ M. Because of the symmetry of
α
,theshape
operator A
ξ
of M is self-adjoint. Its eigenvalues are the principal curvatures of M
with respect to
ξ
. An eigenvector of A
ξ
is called a principal curvature vector of M
with respect to
ξ
, and the eigenvectors corresponding to some principal curvature
form a principal curvature space.Themultiplicity of a principal curvature is the
dimension o f the corresponding principal curvature space.
Since A
s
ξ
X = sA
ξ
X for all s ∈ R, the principal curvatures of M with respect to
s
ξ
are precisely the p rincipal curvatures of M with respect to
ξ
multiplied with the
factor s, and the principal curvature spaces are the same for all 0 = s ∈ R. For this
reason, one is often interested only in the principal curvatures with respect to unit
normal vectors. If, in particular, M is a hypersurface of
¯
M, that is, if the codimension
of M in
¯
M is one, and if
ξ
is a local or global unit normal vector field on M,weoften
speak of the principal curvatures of M without referring to
ξ
. Note that the principal
curvature spaces with respect to linearly independent normal vectors are, in general,
not the same.
Basics of Submanifold Theory in Space Forms 13
We say that a submanifold M of a space form has constant principal curvatures if
for any parallel normal vector field
ξ
along any piecewise differentiable curve c in M
the principal curvatures in direction
ξ
are co nstant functions. We will d eal later with
such submanifolds, starting from Section 3.4. If, in addition, the normal bundle o f M
is flat, we say that the submanifold is isoparametric. The normal bundle, in the very
definition of an isoparametric submanifold, is usually required to be globally flat.
But, a posteriori, this turns out to be equ ivalent to the local flatness of this bundle
(see Page 153).
Since the principal curvatures are roots of a polynomial (namely, the characteris-
tic polynomial of A
ξ
), they are continuous but do not need to be differentiable. For
example, if M is a surface in R
3
, since the principal curvatures
λ
1
and
λ
2
can be ex-
pressed in terms of the Gaussian curvature K and the (length of the) mean curvature
H by
λ
i
= H ±
H
2
−K,
it is clear that they are differentiable o n the set of nonumbilical points, that is, where
λ
1
=
λ
2
. A simple example of a surface where the principal curvatures are not smooth
is the monkey saddle
z =
x
3
−3xy
2
3
.
Here, the principal curvatures are not smooth in the origin; see, for example, [73].
However, if the multiplicities of the principal curvatures are constant on the unit
normal bundle, then the principal curvatures are smooth functions.
1.3.2 Principal curvature distributions and nullity
Let
ξ
be a local unit normal vector field on M that is defined on a connected open
subset U of M.ThenA
ξ
is smoothly diagonalizable over an open and dense subset of
U. On each connected component of this subset we have k smooth eigenvalue func-
tions
λ
i
with multiplicities m
i
, m
1
+ ...+ m
k
= m = dimM. The principal curvature
space with respect to
λ
i
is
E
λ
i
= E
i
= ker{A
ξ
−
λ
i
id}.
We also call E
i
a curvature distribution. Note that if
ξ
happens to be a global unit
normal vector field on M and the principal curvatures of M are constant with respect
to
ξ
, then each curvature distribution is globally defined on M.AcurveinM,allof
whose tangent vectors belong to a curvature distribution, is called a curvature line of
M. Some curvature lines on the monkey saddle are illustrated in Figure 1.1.
More generally, a curvature surface is a connected submanifold S of M for which
there exists a parallel unit normal vector field
ξ
such that T
p
S is contained in a prin-
cipal curvature space of the shape operator A
ξ
p
for all p ∈ S.
A submanifold M in R
n
or S
n
is said to be a Dupin submanifold if the princ ipal
curvatures are constant along all curvature surfaces of M. A Dupin submanifold is
called proper if the number g of distinct principal curvatures of A
ξ
is constant on
the unit normal bundle of M. Important examples of Dupin submanifolds are Dupin
14 Submanifolds and Holonomy
FIGURE 1.1: Principal curvature lines on the monkey saddle z =(x
3
−3xy
2
)/3.
cyclides (Figure 1.2) and isoparametric submanifolds. Dupin cyclides can be charac-
terized by the property that their curvature lines are circles or straight lines. We will
not study Dupin submanifolds in this book and refer to [66, 67, 72] for more details.
The linear subspace
E
0
(p)=
ξ
∈
ν
p
M
ker A
ξ
of T
p
M is called the nullity space of M at p. The collection E
0
of all these spaces is
called the nullity distribution on M. Note that this is actually a distribution only on
any connected component of a suitable dense and open subset of M.
FIGURE 1. 2: A Dupin cyclide.
Basics of Submanifold Theory in Space Forms 15
1.4 Totally geodesic submanifolds of space forms
Let M be a submanifold of a Riemannian manifold
¯
M and
γ
be a geodesic in
M. Then the Gauss formula says that
α
(
˙
γ
,
˙
γ
) is the second derivative of
γ
when
considered as a curve in the ambient space
¯
M.Since
2
α
(v, w)=
α
(v + w, v + w) −
α
(v, v) −
α
(w,w)
for all v, w ∈ T
p
M, p ∈ M, we see that the second fundamental form
α
vanishes
precisely if every geodesic in M is also a geodesic in
¯
M. In such a case, M is called a
totally geodesic submanifold of
¯
M. The basic problems concerning totally geodesic
submanifolds deal with existence, classification, and congruency. We will discuss this
more thoroughly in the general context of submanifolds of Riemannian manifolds in
Section 10.3.
For submanifolds of space forms, we have a positive answer regarding the exis-
tence problem in the following sense. For each point p ∈
¯
M and every linear sub-
space V of T
p
¯
M there exists a totally geodesic submanifold M of
¯
M with p ∈ M and
T
p
M = V . Moreover, since the exponential map exp
p
: T
p
¯
M →
¯
M maps straight lines
through the origin 0 ∈T
p
¯
M to geodesics in
¯
M, there is an open neighborhood U of 0
in T
p
¯
M such that exp
p
maps U ∩V diffeomorphically onto some open neighborhood
of p in M.ThisimpliesthatM is uniquely determined near p, and that any totally
geodesic submanifold of
¯
M containing p and being tangent to V is contained as an
open part in a maximal one with this property. This feature is known as rigidity of
totally geodesic submanifolds.
Geodesics are clearly the simplest examples o f totally geodesic submanifolds.
In the standard space form models
¯
M
n
(
κ
),asdiscussedinSection1.2,wehavean
explicit description of geodesics. Let p ∈
¯
M
n
(
κ
) and v ∈ T
p
¯
M
n
(
κ
). The geodesic
γ
v
: R →
¯
M
n
(
κ
) with
γ
v
(0)=p and
˙
γ
v
(0)=v is given by
γ
v
(t)=
⎧
⎪
⎨
⎪
⎩
cos(
√
κ
t)p +
1
√
κ
sin(
√
κ
t)v if
κ
> 0,
p + tv if
κ
= 0,
cosh(
√
−
κ
t)p +
1
√
−
κ
sinh(
√
−
κ
t)v if
κ
< 0.
This gives an explicit classification of the one-dimensional totally geodesic subman-
ifolds of
¯
M
n
(
κ
).
From this we also easily see that the canonical embeddings
¯
M
k
(
κ
) ⊂
¯
M
n
(
κ
),
1 < k < n, are totally geodesic. The isometry group of
¯
M
n
(
κ
) acts transitively on the
pairs (p,V ) with p ∈
¯
M
n
(
κ
) and V a k-dimensional linear subspace of T
p
¯
M
n
(
κ
).This,
together with the uniqueness properties described above, establishes the classification
of the totally geodesic submanifolds in the standard space forms:
Theorem 1.4.1 Let p ∈
¯
M
n
(
κ
) and V be a k-dimensional linear subspace of
T
p
¯
M
n
(
κ
), 0 < k < n. Then there exists a connected, complete, totally geodesic sub-
manifold M of
¯
M
n
(
κ
) with p ∈ M and T
p
M = V . Moreover, M is congruent to the
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