Basics of Submanifold Theory in Space Forms 21
If
κ
= 0, we consider the vector field Y on M defined by
Y
p
= p +
1
H
ξ
p
for all p ∈ M.ForanyX ∈ TM,wethenget
¯
∇
X
Y = X −
1
H
A
ξ
X = X −X = 0.
So Y is a constant vector field, say Y = p
o
, and therefore M is contained in the
hypersphere with radius
1
H
and center p
o
. Moreover, since H is orthogonal to this
hypersphere, M is minimal in it.
If
κ
= 0, we regard
¯
M
n
(
κ
) as a hypersurface in R
n+1
resp. R
n,1
, and similar
arguments imply the result.
1.7 Reducibility of submanifolds
1.7.1 Submanifold products and extrinsically reducible submanifolds
Let M
1
,...,M
s
,
¯
M
1
,...,
¯
M
s
be Riemannian manifolds and f
i
: M
i
→
¯
M
i
, i =
1,...,s, be isometric immersions. The product map
f = f
1
×...× f
s
: M
1
×...×M
s
→
¯
M
1
×...×
¯
M
s
, (p
1
,...,p
s
) →( f
1
(p
1
),..., f
s
(p
s
))
is called the immersion product of f
1
,..., f
s
or the submanifold product of M
1
,...,M
s
in
¯
M
1
×...×
¯
M
s
. There are simple equations relating the second fundamental form
and the mean curvature vector field of a submanifold product with those o f its factors.
Recall that there is a natural isomorphism
T
(p
1
,...,p
s
)
(M
1
×...×M
s
)
∼
=
T
p
1
M
1
⊕...⊕T
p
s
M
s
,
which we will use frequently below. Denote by
α
i
the second fundamental form and
by H
i
the mean curvature vector field of M
i
. Then the second fundamental form
α
of
M
1
×...×M
s
is given by
α
((X
1
,...,X
s
),(Y
1
,...,Y
s
)) = ((
α
1
(X
1
,Y
1
),...,
α
s
(X
s
,Y
s
))
for all X
i
,Y
i
∈T
p
i
M
i
. Similarly, the mean curvature vector field H of M
1
×...×M
s
is
given by
H =(H
1
,...,H
s
).
More generally, let M = M
1
×...×M
s
be a submanifold o f a Riemannian mani-
fold
¯
M, where dimM
i
≥ 1foralli = 1,...,s and s ≥ 2. Here,
¯
M is not necessarily
a Riemannian product. We denote by L
1
,...,L
s
the totally geodesic foliations on M
22 Submanifolds and Holonomy
that are canonically induced by the product structure of M. For instance, the leaf
L
1
(p) of L
1
through p =(p
1
,...,p
s
) is M
1
×{p
2
}×...×{p
s
}. Note that H
i
= TL
i
is a parallel distribution on M for each i ∈{1,...,s}. We say that M is extrinsically
reducible in
¯
M,orM is an extrinsic product in
¯
M, if the second fundamental form
α
of M satisfies
α
(X
i
,Y
j
)=0forallX
i
∈ T
p
L
i
(p),Y
j
∈ T
p
L
j
(p),i = j, p ∈ M.
From the above equation for the second fundamental form of submanifold products
we immediately see that each submanifold p roduct M in a Riemannian product ma-
nifold
¯
M is extrinsically reducible in
¯
M.
We say that a submanifold M is locally extrinsically reducible in
¯
Matp∈ M if
there exists an open neighborhood of p in M that is extrinsically reducible in
¯
M.Fi-
nally, we say that M is locally extrinsically reducible in
¯
M if M is locally extrinsically
reducible in
¯
M at each point p ∈ M.
1.7.2 Extrinsically reducible submanifolds of R
n
and S
n
There is a useful criterion for local extrinsic reducibility of submanifolds in Eu-
clidean spaces due to Moore [218].
Lemma 1.7.1 (Lemma of Moore) Let M be a submanifold of R
n
. If there exists a
nontrivial parallel distribution H on M such that the second fundamental form
α
of
M satisfies
α
(H ,H
⊥
)=0,
then M is locally a submanifold product in R
n
and hence locally extrinsically re-
ducible in R
n
. Moreover, if f : M → R
n
is a complete simply connected immersed
submanifold, then it is a n immersion product.
Proof Since H is a parallel distribution on M, H
⊥
is also a parallel distribution on
M. Hence, both H and H
⊥
are integrable with totally geodesic leaves. We choose
and fix a point p ∈ M. By the de Rham Decomposition Theorem there exists an
open neighborhood of p in M that is isometric to the Riemannian product M
1
×M
2
,
where M
1
and M
2
are connected integral manifolds of H and H
⊥
containing p,
respectively. We will now prove that M
1
×M
2
is a submanifold product in R
n
.
For each point q =(q
1
,q
2
) ∈ M
1
× M
2
we define L
1
(q
1
)={q
1
}×M
2
and
L
2
(q
2
)=M
1
×{q
2
}. We now choose two points q =(q
1
,q
2
) and ˜q =(˜q
1
, ˜q
2
) in
M
1
×M
2
and two tangent vectors X ∈ T
q
L
1
(q
1
)=H
⊥
q
and Y ∈T
˜q
L
2
( ˜q
2
)=H
˜q
.Let
c : [0, 1] → L
2
(q
2
) be a smooth curve with c(0)=q =(q
1
,q
2
) and c(1)=(˜q
1
,q
2
).
Let E
X
be the ∇-parallel vector field along c with E
X
(0)=X,where∇ is the Levi-
Civita connection of M. By construction, ˙c is tang ent to H everywhere, and since
X ∈ H
⊥
and H
⊥
is a parallel distribution on M, we see that E
X
is tangent to H
⊥
everywhere. Since
α
(H ,H
⊥
)=0 by assumption, the Gauss formula for M ⊂ R
n
implies
¯
∇
˙c
E
X
= ∇
˙c
E
X
+
α
( ˙c,E
X
)=0.
Basics of Submanifold Theory in Space Forms 23
Thus, E
X
is a
¯
∇-parallel vector field along c and hence E
X
(t)=X for all t ∈ [0,1],
where we identify, as usual, the tangent spaces of R
n
with R
n
in the canonical way. It
follows that E
X
(1)=X ∈ H
⊥
. Next, let d : [0,1] → L
1
( ˜q
1
) be a smooth curve with
d(0)= ˜q =(˜q
1
, ˜q
2
) and d(1)=(˜q
1
,q
2
),andletE
Y
be the ∇-parallel vector field along
d with E
Y
(0)=Y . As above we can show that E
Y
(1)=Y ∈H . We thus have proved
that T
q
L
1
(q
1
) and T
˜q
L
2
( ˜q
2
) are perpendicular to each other for all q, ˜q ∈ M
1
×M
2
.
Since R
n
is homogeneous, we can assume without loss of generality that p is
the origin of R
n
.LetR
n
1
and R
n
2
be the linear subspaces of R
n
that are g enerated
by the linear subspaces T
q
L
2
(q
2
)=H
q
and T
q
L
1
(q
1
)=H
⊥
q
for all q ∈ M
1
×M
2
,
respectively. We have just proved that R
n
1
and R
n
2
are perpendicular to each other.
By construction, we have M
1
×M
2
⊂ R
n
1
×R
n
2
, which shows that M
1
×M
2
is a
submanifold product in R
n
.
The global version follows with the same arguments by using the global de Rham
Decomposition Theorem.
Since S
n
is a totally umbilical sub manifold of R
n+1
, the Lemma of Moore im-
plies:
Corollary 1.7.2 Let M be a submanifold of S
n
and consider S
n
as a submanifold of
R
n+1
. If there exists a nontrivial parallel distribution H on M such that the second
fundamental form
α
of M in S
n
satisfies
α
(H ,H
⊥
)=0,
then M is locally a submanifold product in R
n+1
and hence locally extrinsically re-
ducible in S
n
.
1.7.3 Extrinsically reducible submanifolds of R
n,1
and H
n
The Lorentzian analogue of the Lemma of Moore is not a straightforward gen-
eralization. This is due to the fact that, in Lorentzian spaces, there exist degenerate
linear subspaces. Recall that a linear subspace V of R
n,1
is degenerate if there exists
a nonzero vector v ∈V such that v,w = 0forallw ∈V . Evidently, any such vector
v must be lightlike.
Proposition 1.7.3 (Lorentzian version of Lemma of Moore) Let M be a Riemann-
ian submanifold of R
n,1
. If there exists a nontrivial parallel distribution H on M
such that the linear subspaces H
q
,q∈ M, span a nondegenerate linear subspace V
of R
n,1
and such that the second fundamental form
α
of M satisfies
α
(H ,H
⊥
)=0,
then M is locally a submanifold product in R
n,1
and hence locally extrinsically re-
ducible in R
n,1
.
Proof The first p art of the proof is analogous to the one in the Euclidean case. (Since
we assume M to be a Riemannian submanifold we can apply the de Rham Decom-
position Theorem.) Since V is nondegenerate by assumption, it is either a Lorentzian
24 Submanifolds and Holonomy
or a Euclidean subspace of R
n,1
and hence V
⊥
is either Euclidean or Lorentzian, re-
spectively. It follows that M
1
×M
2
⊂V ×V
⊥
⊂ R
n,1
, which shows that M is locally
a submanifold product in R
n,1
.
Using the Lorentzian version of Moore’s lemma, we now derive a reducibility
result for submanifolds in real hyperbolic spaces.
Corollary 1.7.4 Let M be a submanifold of H
n
and consider H
n
as a submanifold
of R
n,1
. If M is not contained in a horosphere of H
n
and if there exists a nontrivial
parallel distribution H on M such that the second fundamental form
α
of M ⊂ H
n
satisfies
α
(H ,H
⊥
)=0,
then M is locally a submanifold product in R
n,1
and hence locally extrinsically re-
ducible in H
n
.
Proof Since H
n
is totally umbilical in R
n,1
,wealsohave
˜
α
(H ,H
⊥
)=0, where
˜
α
is the second fundamental form of M in R
n,1
. Moreover, since M is a submanifold
of H
n
and H
n
is a Riemannian submanifold of R
n,1
, M is a Riemannian submanifold
of R
n,1
.LetV
1
and V
2
be the span of the linear subspaces H
q
and H
⊥
q
, q ∈ M,
respectively. It remains to show that either V
1
or V
2
is a nondegenerate subspace of
R
n,1
. Assume that both V
1
and V
2
are degenerate. Then, since V
1
is orthogonal to
V
2
,alsoV
1
+V
2
is a degenerate subspace of R
n,1
. It follows that M is contained in an
affine subspace of R
n,1
whose linear part is degenerate. Since the intersection of such
an affine subspace with H
n
is a horosphere in H
n
, we see that M lies in a horosphere
of RH
n
, which is a contradiction.
Remark 1.7.5 Any horosphere in H
n
is isometric to the Euclidean space R
n−1
. Con-
sequently, a submanifold of H
n
that is contained in a horosphere can be regarded as
a submanifold of R
n−1
.
1.8 Exercises
Exercise 1.8.1 Let M be a hypersurface of R
n
with m = dimM = n −1 > 2. Let
κ
1
,...,
κ
m
be the principal curvature functions of M. Assume that at least three prin-
cipal curvatures are nonzero, that is, the rank of the shape operator is at least 3 at each
point of M. Prove th at the sectional cur vatures of M determine
κ
1
,...,
κ
m
. Deduce
the Beez-Killing Theorem, which states that the second fundamental form of M is
determined by the first fundamental form of M (cf. [19], 10.8).
Exercise 1.8.2 Let f : S
2
→ R
5
be given by
(x,y,z) →
xy,xz,yz,
1
2
(x
2
−y
2
),
1
2
√
3
(x
2
+ y
2
−2z
2
)
.
Basics of Submanifold Theory in Space Forms 25
(a) Verify that f induces an embedding
˜
f of the real projective plane RP
2
into the
hypersphere S
4
(
1
√
3
) of R
5
.
(b) Compute the second fundamental form of f (or of
˜
f ), verifying that they are
minimal in the sphere.
˜
f is called the Veronese surface.
Exercise 1.8.3 Let P : S
1
(R) ×S
1
(R) → R
4
be the Clifford torus, which is given by
(u,v) → (R cos(u),Rsin(u),Rcos(v),R sin(v)) .
Compute the second fundamental form of P.
Exercise 1.8.4 Give a direct proof of the fact that the totally geodesic submanifolds
of R
n
are affine subspaces. [Hint: See the proof of Theorem 1.5.1 (Reduction of
codimension).]
Exercise 1.8.5 (cf. [317], Corollary 1.5) Let M be a submanifold of a space form
and assume that there exists a parallel normal vector field
ξ
on M such that the
eigenvalues o f A
ξ
are all distinct. Prove that M has flat normal bundle.
Exercise 1.8.6 Prove that the connected, complete, totally geodesic (resp. totally
umbilical) submanifolds M of S
n
(r) ⊂R
n+1
with m = dimM ≥2 are the intersections
of S
n
(r) with the (m + 1)-dimensional linear (resp. affine) subspaces of R
n+1
.
Exercise 1.8.7 Prove that the connected, complete, totally geodesic (resp. totally
umbilical) submanifolds M of H
n
(r) ⊂R
n,1
with m = dim M ≥2 are the intersections
of H
n
(r) with the (m + 1)-dimensional linear (resp. affine) subspaces of R
n,1
.
Exercise 1.8.8 Prove that a submanifold M of R
n
with parallel second fundamental
form has parallel first normal space.
Exercise 1.8.9 Prove that two autoparallel distributions that are orthogonally com-
plementary are both parallel. Is this result still true for three autoparallel distribu-
tions?
Exercise 1.8.10 Let M be a submanifold of R
n
with parallel second fundamental
form and assume that the shape operator A
H
with respect to the mean curvature vector
field H has at least two distinct eigenvalues. Prove that M is locally re ducible.
Exercise 1.8.11 Let M be a totally geodesic submanifold o f a space form and let N
be a submanifold of M. Prove that
A
M
ξ
X = A
N
ξ
X for all
ξ
∈
ν
p
M,X ∈T
p
N, p ∈ N,
where A
M
and A
N
are the shape operators of M and N, respectively. Conversely,
prove that if the above property holds for any submanifold N of M,thenM is a
totally umbilical submanifold.
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