16 Submanifolds and Holonomy
canonical totally geodesic embedding of
¯
M
k
(
κ
) in
¯
M
n
(
κ
). Each connected, totally
geodesic submanifold N of
¯
M
n
(
κ
) with p ∈ N and T
p
N = V is an open part o f M.
Actually, it is not difficult to show d irectly that the totally geodesic submanifolds
of R
n
are the affine subspaces (see Exercise 1.8.4). Moreover, the connected, com-
plete, totally geodesic submanifolds of S
n
(r) ⊂ R
n+1
are precisely the intersections
of S
n
(r) with the linear subspaces of R
n+1
. Analogously, the connected, complete, to-
tally geodesic submanifolds of H
n
(r) ⊂ R
n,1
are precisely the intersections of H
n
(r)
with the linear Lorentzian subspaces of R
n,1
. We also propose as an exercise to give
a d irect proof of this (see Exercises 1.8.6 and 1.8.7).
1.5 Reduction of the codimension
A submanifold M of a Riemannian manifold
¯
M is said to be a full submanifold if
it is not contained in any totally geodesic submanifold N of
¯
M with dimN < dim
¯
M.
If M is not full in
¯
M, we say that there is a reduction of the codimension of M.For
example, M is full in R
n
if and only if M is not contained in any affine hyperplane of
R
n
.IfM is not full in R
n
, then there exists a smallest affine subspace of R
n
containing
M, namely the intersection o f all affine subspaces containing M.Ifk is the dimension
of this affine subspace, then we might view M as a full submanifold of R
k
.This
means that we have reduced the codimension of M by n −k dimensions.
In order to reduce the codimension, it is useful to study a particular subspace of
the normal space called the first normal space.Thefirst normal space of M at p ∈ M
is defined as the linear subspace N
1
p
of
ν
p
M spanned by the image of the second
fundamental form at p,thatis,
N
1
p
= span of {
α
(v, w) ∈
ν
p
M : v,w ∈ T
p
M} =
ν
p
M {
ξ
∈
ν
p
M : A
ξ
= 0},
where the last set is the orthogonal complement in
ν
p
M of the linear subspace of
ν
p
M
consisting of all normal vectors
ξ
at p for which the shape operator A
ξ
vanishes. If
the dimension of the first normal space does not depend on p,thenN
1
is a smooth
subbundle of the normal bundle
ν
M.
Let N be a subbundle of
ν
M. We say that N contains the first normal bundle
if N
1
p
⊂ N
p
for all p ∈ M (we do not assume here that the first normal bundle is
smooth). The subbundle N is said to be a parallel subbundle if
∇
⊥
v
ξ
∈ N for all sections
ξ
of N and all v ∈ TM.
Equivalently, for any curve c : [a, b] → M we have
τ
⊥
c
(N
c(a)
)=N
c(b)
,where
τ
⊥
c
is
the ∇
⊥
-parallel transport along c. Recall that the rank of N is the dimension of any
of its fibers.
The following criterion is very useful in this context of submanifold theory in
space forms (see [97, Chapter 4], or [121]).
Basics of Submanifold Theory in Space Forms 17
Theorem 1.5.1 (Reduction of codimension) Let f : M →
¯
M be an isometric im-
mersion from an m-dimensional connected Riemannian manifold M into an n-
dimensional standard space form
¯
M . Assume that there exists a parallel subbundle
N of the normal bundle
ν
M with k = rk(N ) that contains the first normal space
of M. Then there exists an (m + k)-dimensional totally geodesic submanifold N of
¯
M
such that f is an isometric immersion from M into N.
Proof Since N is a parallel subbundle of
ν
M, its orthogonal complement N
⊥
=
ν
M N in
ν
M is also a parallel subbundle of
ν
M. Moreover, since N contains the
first normal space, we have A
ξ
= 0forall
ξ
∈ N
⊥
. We divide the proof into three
separate cases according to the sign of the curvature of
¯
M. It is also clear that we
can restrict to the model spaces R
n
, S
n
,andH
n
because a homothetic change of the
metric does not affect the assertion.
(1) The case
¯
M = R
n
.
Let p ∈ M and c be a curve in M with c(0)=p.Let
ξ
0
∈ N
⊥
p
and
ξ
be the parallel normal vector field along c with
ξ
(0)=
ξ
0
.Since
N
⊥
is invariant under ∇
⊥
-parallel translation, the Weingarten formula
¯
∇
˙c
ξ
= −A
ξ
˙c + ∇
⊥
˙c
ξ
= 0
implies that
ξ
is a
¯
∇-parallel vector field along c. Since the parallel tra nsport in R
n
along curves is independent of initial and end points of the cu rve, this implies that
the parallel transpor t of
ξ
0
is independent of the curve. Hence, for any
ξ
0
∈ N
⊥
p
the
parallel transport of
ξ
0
along curves determines a well-defined parallel normal vector
field
ξ
on M. Thus, there exists a parallel orthonormal frame field
ξ
1
,...,
ξ
n−m−k
of
N
⊥
. As we have just seen, each normal vector field
ξ
i
is the restriction to M of
a suitable constant vector field on R
n
, which we also denote by
ξ
i
. For each i ∈
{1,...,n −m −k}we define the height function
f
i
: M → R , p →f (p),
ξ
i
.
For all v ∈ T
p
M and p ∈ M we then get
d
p
f
i
(v)=d
p
f (v),
ξ
i
(p)−f (p),A
ξ
i
(p)
v = 0.
Thus, f
i
is constant (since M is con nected) and it follows that f (M) is contained in
the intersection of n −m −k affine hyperplanes of R
n
with pairwise linearly indepen-
dent normal directions. Such an intersection is isometrically congruent to the totally
geodesic R
m+k
⊂ R
n
, by which the assertion is proved.
(2) The case
¯
M = S
n
.
Consider S
n
as the unit sphere in R
n+1
with center at the
origin. Let
ζ
be the unit normal vector field on S
n
in R
n+1
pointing outward, that is,
ζ
p
= p for all p ∈S
n
. Recall that the Levi-Civita connection
¯
∇ of S
n
is the orthogonal
projection onto the tangent spaces of S
n
of the directional derivative D of R
n+1
.Then
we get
D
v
ζ
= v and D
v
ξ
=
¯
∇
v
ξ
for all v ∈ TM and all normal vector fields
ξ
on M, which are tangent to S
n
. Thus,
when we consider f as an isometric immersion into R
n+1
, the bundle N may be
18 Submanifolds and Holonomy
regarded as a parallel subbundle of the normal bundle of M in R
n+1
. The subbundle
¯
N = N ⊕R
ζ
is a parallel subbundle of the normal bundle of M regarded as a sub-
manifold of R
n+1
. Moreover,
¯
N contains the first normal space of the submanifold
M of R
n+1
. We also see that
¯
N
1
is invariant under D
⊥
-parallel transport, where D
⊥
is the normal connection of M regarded as a submanifold o f R
n+1
. By case (1) we
now see that f is an isometric immersion into some totally geodesic R
m+k+1
⊂R
n+1
.
But, since R
m+k+1
contains R
ζ
, it also contains the origin of R
n+1
, and it follows
that f is an isometric immersion into the totally geodesic R
m+k+1
∩S
n
= S
m+k
.
(3) The case
¯
M = H
n
.
The p roof is similar to case (2) and we therefore sketch it
here only. We consider H
n
as a hypersurface in the Lorentzian space R
n,1
and denote
by
ζ
the timelike unit normal vector field on H
n
given by
ζ
p
= p for all p ∈ H
n
.As
in case (2) we define the subbundle
¯
N of the the normal bundle of M,regardedas
a Riemannian submanifold of the Lorentzian space. We then prove, as in case (1),
that f is an isometric immersion into an affine subspace of R
n,1
whose linear part is
a Lorentzian subspace of R
n,1
.Thisaffine subspace then contains the origin of R
n,1
,
which implies that f is an isometric immersion into a totally geodesic H
m+k
⊂ H
n
.
Some necessary and sufficient conditions for the invariance of the first normal
bundle under parallel transport were obtained by do Carmo, Colares, Dajczer, and
Rodriguez and can be found in [97, Section 4.2]. A certain generalization to arbitrary
Riemannian manifolds
¯
M can be found in [283]. Theorem 1.5.1 was generalized by
Di Scala and Vittone [110] to Riemannian symmetric spaces
¯
M.
Remark 1.5.2 If a submanifold M of Euclidean or Lorentzian space admits a
nonzero p arallel normal vector field
ξ
such that A
ξ
= 0, then M is not full (Exer-
cise: What happens in the case of a submanifold of the sphere?).
1.6 Totally umbilical submanifolds of space forms
1.6.1 Totally umbilical submanifolds and extrinsic spheres
Let M be a submanifold of a Riemannian manifold
¯
M.Ifp ∈ M,
ξ
∈
ν
p
M and
A
ξ
=
ρ
id
T
p
M
for some
ρ
∈R,thenM is said to be umb ilical in direction
ξ
. A normal
vector field
ξ
on M such that A
ξ
=
ρ
id
M
for some smooth function
ρ
on M is called
an umbilical normal vector field or umbilical normal section of M.IfM is umbilical
in any no rmal direction
ξ
,thenM is called a totally umbilical submanifold of
¯
M.A
submanifold M is totally umbilical if and only if
α
(X,Y )=X,Y H
for all vector fields X,Y on M,whereH is the mean curvature vector field on M.It
is obvious that every one-dimensional submanifold and every totally geodesic sub-
Basics of Submanifold Theory in Space Forms 19
manifold is totally umbilical. It is also clear that conformal transformations of
¯
M
preserve totally umbilical submanifolds.
A totally umbilical submanifold with nonzero parallel mean curvature vector field
is called an extrinsic sphere. I n a sp ace form the two concepts of totally umbilical
(and non-totally geodesic) submanifolds and extrinsic spheres coincide in dimen-
sions ≥ 2. Indeed, we have the following:
Lemma 1.6.1 Let M be a totally umbilical submanifold of a space form with
dimM ≥ 2. Then we have
∇
⊥
H = 0 and R
⊥
= 0.
In particular, M is an extrinsic sphere.
Proof Since M is totally umbilical, the shape operator A of M satisfies
A
ξ
X =
ξ
,HX
for all normal vector fields
ξ
and tangent vector fields X on M. The Ricci equation
then implies R
⊥
= 0 (see also Exercise 1.8.12). Moreover, since the second funda-
mental form of M is of the form
α
(X,Y )=X,Y H, the Codazzi equation implies
Y,Z∇
⊥
X
H = X ,Z∇
⊥
Y
H.
Since dim M ≥ 2, we can choose X and Y = Z to be orthonormal, which implies
∇
⊥
H = 0.
The connected, complete, totally umbilical and non-totally geodesic submani-
folds M with m = dimM ≥ 2ofR
n
, S
n
and H
n
are as follows (see also Exercises
1.8.6 and 1.8.7):
1. In R
n
: M is a sphere S
m
(r) with some r ∈R
+
.
2. In S
n
: M is an m-dimensional sphere which is obtained by intersecting S
n
with
an affine and nonlinear (m + 1)-dimensional subspace of R
n+1
.
3. In H
n
: We consider H
n
⊂R
n,1
.ThenM is obtained by intersecting H
n
with an
affine and nonlinear (m + 1)-dimensional subspace of R
n,1
.
In particular, the totally umbilical hypersurfaces in H
n
are the intersections of H
n
with the affine hyperplanes of R
n,1
whose vector part is (Ra)
⊥
for some a ∈ R
n,1
.
Moreover:
(a) If a is a timelike vector in R
n,1
, in which case (Ra)
⊥
is a Euclidean vector
space, the totally umbilical hypersurfaces obtained in this way are g eodesic
hyperspheres. A geodesic hypersphere M
r
(p) in H
n
is the set of all points in
H
n
with distance r > 0 to a point p ∈ H
n
.
(b) If a is a spacelike vector in R
n,1
, in which case (Ra)
⊥
is a Lorentzian vector
space, the totally umbilical hypersurfaces obtained in this way are the hyper-
surfaces that are equidistant to a totally geodesic H
n−1
⊂ H
n
.
20 Submanifolds and Holonomy
(c) If a is a lightlike vector in R
n,1
, in which case (Ra)
⊥
is a degenerate vector
space, the totally umbilical hypersurfaces obtained in this way are horospheres.
In the Poincar´e disk model of H
n
, the horospheres are the spheres in the ball that
are tangent to the boundary sphere of the ball. In this model, it is clear that horo-
spheres are totally umbilical. Indeed, the identity map from the ball equipped with
the Euclidean metric onto the ball equipped with the Poincar´e metric is a conformal
transformation. Therefore, it sends the spheres tangent to the boundary sphere of the
ball, which are totally umbilical, onto totally umbilical submanifolds of H
n
.
In the half space model, the hypersurfaces x
n
= c, c > 0, are horospheres. Actu-
ally, x
n
= c gives a family of parallel hypersurfaces that are all centered at the same
point at infinity. Moreover, in this model, it is easy to see that every horosphere in H
n
is isometric to the Euclidean space R
n−1
, and that they are totally umbilical, b ecause
of the description of the geodesics in this model.
We can summarize the above discussion on extrinsic spheres in space forms in
the f ollowing th eorem, which gives us an explicit description.
Theorem 1.6.2 Let p ∈
¯
M
n
(
κ
),n≥ 3, V be a linear subspace of T
p
¯
M
n
(
κ
) with
1 < dimV < n, and let 0 =
ξ
∈ T
p
¯
M
n
(
κ
) be orthogonal to V . Then there exists a
unique connected complete extrinsic sphere M of
¯
M
n
(
κ
) with p ∈ M, T
p
M = V and
such that the mean curvature vector field H of M satisfies H
p
=
ξ
. Moreover, M is a
space of constant curvature
κ
+
ξ
,
ξ
.
A survey about totally umbilical submanifolds in more general ambient spaces,
as well as many references, can be found in [78]. We discuss totally umbilical sub-
manifolds and extrinsic spheres in symmetric spaces in Section 11.2.
1.6.2 Pseudoumbilical submanifolds
A generalization of totally umbilical submanifolds is that of pseudoumbilical
submanifolds. A submanifold M of a Riemannian manifold
¯
M is called pseudoum-
bilical if M is umbilical in the direction of the mean curvature vector field H every-
where. This just means
α
(X,Y ),H = X ,YH
2
for all X,Y ∈ T
p
M, p ∈ M. Chen and Yano proved in [82] the following result:
Proposition 1.6.3 (Chen, Yano) Let M be a pseudoumbilical submanifold of a stan-
dard space form
¯
M
n
(
κ
). If the mean curvature vector field H of M is parallel, then
(a) M is a minimal submanifold of
¯
M
n
(
κ
),or
(b) M is a minimal submanifold of some extrinsic sphere in
¯
M
n
(
κ
).
Proof By assumption, H is constant. If H = 0, then M is a minimal submanifold
of
¯
M
n
(
κ
). Let us assume that H = 0. Then
ξ
=
H
H
is a ∇
⊥
-parallel unit normal
vector field on M.
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