Classification of totally umbilical submanifolds in symmetric space

Abstract

A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is no totally umbilical submanifold of codimension less than rank M — 1 in any irreducible symmetric space M. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some classification theorems of such submanifolds are obtained.

Full text

J. Austral. Math. Soc. (Series A)
M>
(1980), 129-136
CLASSIFICATION OF TOTALLY UMBILICAL
SUBMANIFOLDS IN SYMMETRIC SPACES
BANG-YEN CHEN
(Received 31 January 1979; revised 28 April 1980)
Communicated by J. Virsik
Abstract
A submanifold of a Riemannian manifold is called a totally umbilical submanifold if the second
fundamental form is proportional to the first fundamental form. In this paper, we shall prove that there is
no totally umbilical submanifold of codimension less than rank M—
1
in any irreducible symmetric space
Vf. Totally umbilical submanifolds of higher codimensions in a symmetric space are also studied. Some
classification theorems of such submanifolds are obtained.
1980 Mathematics subject classification [Amer. Math. Soc): 53 B 20, 53 C 35, 53 C 40.
1.
Introduction
Let N be an n-dimensional submanifold of an m-dimensional Riemannian manifold
M (n ^ 2) with metric g. Let V and
V
be the covariant differentiations on N and M,
respectively. Then the second fundamental form h of the immersion is defined by the
equation
(1.1)
h(X,Y)
=
WxY~VxY,
where X and 7are vector fields tangent to N. The submanifold N is said to be totally
umbilical if
(1.2)
h(X,Y)
=
g(X,Y)-H,
for all vector fields X, Ytangent to N, where H = 1/n trace h is the mean curvature
vector of N in M. The length of H is called the mean curvature of N in M. A totally
umbilical submanifold with vanishing mean curvature is called a totally geodesic
submanifold.
© Copyright Australian Mathematical Society 1980
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129

130 Bang-Yen Chen [2]
In Wolf (1963), totally geodesic submanifolds in rank one symmetric spaces are
classified. Totally geodesic submanifolds in symmetric spaces of higher ranks have
been studied in Chen-Nagano (1977, 1978) in which the (M + ,M_)-theory is
introduced. In this paper, we shall generalize the results of Wolf (1963) and Chen-
Nagano (1977, 1978) to totally umbilical submanifolds. In particular, we shall prove
the following theorems.
THEOREM 1. If an irreducible symmetric space M admits a totally umbilical
hypersurface N, then both M and N are of constant curvature.
THEOREM 2. Let N be a totally umbilical submanifold in a symmetric space M. If
dimM
—
dimAf <rankM— 1, then the mean curvature vector is parallel (in the
normal bundle). In particular, the mean curvature is constant and N is either totally
geodesic or of constant curvature.
THEOREM
3.
IfN
is a
totally
umbilical
submanifold
in an irreducible
symmetric
space
M, then
(1.3) codimNSs
rankM-1.
THEOREM
4.
Let N be a totally umbilical submanifold with constant
mean
curvature
in a compact symmetric space M. Then
(i) either N is totally geodesic or N is of positive constant curvature, and
(ii) if the mean curvature vector H is not parallel, and dim N > 2, then
dim N <
2
dim M.
REMARK. Totally umbilical submanifold with parallel mean curvature vector in
(locally) symmetric spaces have been completely classified in Chen (1979).
For general results on symmetric spaces, see Helgason (1968).
The estimate of codimension in Theorem 2 is best possible.
2.
Preliminaries
Let N be an n-dimensional submanifold of a Riemannian manifold M with metric
g. For a vector field £ normal to N, we write
(2.1) Vx^ = -A,
where
—
A^ X, Dx
£,
are the tangential and normal components of Vx £, respectively.
A normal vector field <^issaidtobepara//e/(in the normal bundle) if Dx £ = 0 for all
X tangent to JV.
Let R and R be the curvature tensors associated with V and V, respectively. For

[3] Totally umbilical submanifolds
in
symmetric spaces
131
example,
(2.2) K(.V,n
=
VxVy-VyVx-V[X.y].
A Riemannian manifold
M is
locally symmetric
if VR = 0.
For
the
second fundamental form
h, we
define
the
covariant derivative
in
^),
to be
(2.3)
(V*h)(Y,Z)
=
DMy,2))-h(Wx Y,Z)-h(Y,VxZ),
where
TN and T1 N
denote
the
tangent
and
normal bundles
of N,
respectively.
For vector fields X,Y,Z,W tangent
to N, the
equations
of
Gauss
and
Codazzi
take
the
forms:
(2.4)
K(A\
Y\ Z,
W)
= R(X,
Y;
Z,
W)
+
g{h(X,
W),
h(Y,
Z))-g(h(X,
Z), h(Y,
W)\
(2.5)
(R(X,
Y)Z)1 = (Vxh)(Y,Z)-(Vyh)(X,Z),
where
R(X,
Y;
Z,
W)
=
g{R(X,
Y)Z,
W),...,
etc.,
and l in
(2.5)
denotes
the
normal
component.
Let
A'
and
Y
be
two
orthonormal vectors tangent
to N.
The
sectional curvature
K(X
A
Y) of
the
plane
X
A
Y
spanned
by {X, Y] is
given
by
(2.6)
K(X A
Y)
=
R{X,
Y,
Y,
X).
We denote
by K the
sectional curvature
for M.
By
an
extrinsic sphere
we
mean
a
totally umbilical submanifold with nonzero parallel mean curvature vector.
We
mention
the
following lemma
for
later
use.
LEMMA
1
(Chen 1979)).
Let N be an
n-dimensional extrinsic sphere
in a
locally
symmetric space
M.
Then
M
admits
an
(n+
l)-dimensional totally geodesic
sub-
manifold
N
such that
(a)
N
is
an
extrinsic sphere
of N
and (b)
both
N
and
N
are of
constant curvature.
An isometry
s of a
Riemannian manifold
is
said
to
be
involutive
if
its
iterate
s2 is
the identity map. A Riemannian manifold
M
is
a
symmetric space
if, for
each point
p
of
M,
there exists
an
involutive isometry
s
p
of
M
such that p is
an
isolated fixed point
of
s
p
.
It is
well known that every symmetric space
is a
complete locally symmetric
space.
And
every locally symmetric space
is
locally
an
open submanifold
of a
symmetric space.
We denote
by
G the
closure
of
the
group
of
isometries generated
by
\sp:pe
M)
in
the
compact-open topology. Then
G
acts transitively
on M;
hence
the
typical
isotropy subgroup
X, say at 0, is
compact
and M = G/K.
Let
<70
be
the
involutive automorphism
of
G
given
by a
o
{x)
= s0
•
x
•
s
0
, x
6
G.
Then
ff0
fixes K
and
it
induces
an
involutive automorphism
of
the Lie
algebra
g
of
G.
The

132
Bang-Yen Chen [4]
Cartan decomposition of g is then given by
(2.7) 9=I + m,
where i and m are the eigenspaces of a0 with eigenvalues
1
and -
1,
respectively. It is
known that f is the Lie algebra of K and we have
[I,
f] c f, [f, m] c m, [m, m] c !.
Moreover, m can be identified with the tangent space of M at 0.
The following lemmas of E. Cartan are well known (see Helgason (1968)).
LEMMA
2. The curvature tensor R of M at 0 satisfies
(2.8) R(l,y)Z = -[[A-,y],Z]
for X,Y,Zem.
LEMMA
3.
Let B be a totally geodesic
submanifold
of M through 0. Then B
is
fiat if
and only if
\_n,
7t] = 0 where n = T0B a To M = m.
We mention the following unpublished result of Chen-Nagano for later use. This
result is proved by using the (M
+
,
M_)-theory (Chen-Nagano (1978)).
LEMMA
4. Every totally geodesic
submanifold
B of an
irreducible
locally symmetric
space M satisfies
(2.9) dim B sg dim M - rank M.
3.
Proof of Theorem 1
Let N be a totally umbilical hypersurface of an irreducible symmetric space M.
Then by (1.2) and (2.3) we have
(3.1) {V
Thus by (3.1) and the Codazzi equation, we find
(3.2) R(X, Y; Z,H) =
\{g(Y,
where a2 = g(H, H). Let El,..., £„ be an orthonormal basis of Tp N, pe N. Then (3.2)
implies
(3.3) «(£„£,;£,,//) = \(El a2)

[5] TotalK umbilical submanifolds
in
symmetric spaces
133
for
i =
2,...,
n. Consequently, the Ricci tensor
S of M
satisfies
(3.4) S(£1,//)
=
"^1(£1a2).
On the other hand, since M is Einsteinian, S{E1, H)
=
0.
Thus we find that
El a2 = 0.
Since E{ can be chosen to be any unit vector, a is constant. Moreover, because
N
is a
hypersurface, N is either an extrinsic sphere or a totally geodesic hypersurface. If N is
an extrinsic sphere, Lemma
1
implies that both
N
and M are of constant curvature. If
N is a totally geodesic hypersurface, Lemma 4 implies that
M
is a locally symmetric
space of rank one. From Wolf's result (Wolf(
1963)),
we know that this is impossible
unless M is locally isometric to
a
sphere, or a hyperbolic space. In both cases,
M
has
constant curvature. This proves Theorem
1.
4.
Proof
of
Theorems
2
and
3
Suppose that
N
is
a
totally umbilical submanifold
of a
symmetric space
M.
We
may choose any fixed point p in M as the origin of M. For any given nonzero vector
XeTpN,
there exists
a
flat totally geodesic submanifold
B
through
p
such that
XeTpB and dimB
=
rank M.
If
dim M- dim
N <
rankM- 1, we have
(4.1)
dim{TpN
nTpB)^2.
Thus there
is a
unit vector
Yin TpN
nIpBso that g(X, Y)
=
0. Since
N is
totally
umbilical, we have
(4.2) (R(X,
Y)Z)1 = g(Y,Z)-DxH-g(X,Z)-D\H.
Consequently, we obtain
(4.3) (R(,Y,
Y)Y)L = DXH.
On the other hand, since
B
is
a
flat totally geodesic submanifold
of
M, Lemma
3
says that the tangent space
Tp
B forms an abelian linear subspace of
m.
In particular,
we have [X, Y]
= 0.
Substituting this into (2.8) we find
(4.4) R(X, Y) Y=
0.
Combining (4.3) and (4.4) we find
Dx
H
=
0. Since
X
can be chosen to be any vector
tangent
to N at p
and
p
can be chosen
to be
any point
in N,
the mean curvature
vector
H is
parallel
in the
normal bundle.
In
particular,
the
mean curvature
is
constant and from Lemma 1, we see that
N
is either totally geodesic
or
of constant
curvature. This proves Theorem
2.
If M is an irreducible symmetric space. Lemma 4 shows that
M
admits no totally
geodesic submanifold of codimension
<
rank M. From the discussion above, we see

134 Bang-Yen Chen [6]
that every totally umbilical submanifold of codimension < rank M—
1
is an
extrinsic sphere. Therefore, by using Lemma 1, M admits a totally geodesic
submanifold of codimension < rank M
—2.
This contradicts to Lemma 4 again.
Theorem 3 is thus proved.
5. Proof of Theorem 4
Let N be a totally umbilical submanifold with constant mean curvature in a
symmetric space. Then, by a result of Miyazawa and Chliman (1972), N is either
totally geodesic or a locally symmetric space with vanishing Weyl conformal
curvature tensor. In the following, we assume that N is not totally geodesic.
Case (a). If N is irreducible and locally symmetric, N is Einsteinian. Since every
Einstein manifold with vanishing Weyl conformal curvature tensor is of constant
curvature, N is of constant curvature.
Case
(b).
If N is reducible and of dimension 2 or 3, then N is locally the product of
two locally symmetric spaces of dimension ^ 2. Since every locally symmetric space
of dimension 2 is of constant curvature, N is either flat or a local product of a curve
and a surface of constant curvature.
Case (c). If N is reducible and of dimension > 4, then from Proposition 2 of
Goldberg (1969), we see that N is a flat space, a local product of two spaces of
constant curvatures c and
—
c,
or a local product of a curve and a space of constant
curvature.
Consequently, if N is not totally geodesic, N is one of the following spaces: (1) a
space of constant curvature, (2) a local product of two spaces of constant curvatures
c and —c, respectively, or (3) a local product of a curve and a space of constant
curvature.
Now, if M is a compact symmetric space, M is always nonnegatively curved. Thus,
by the equation of Gauss, if N is not totally geodesic, N is always positively curved.
The cases (2) and (3) cannot occur. Therefore, N is either totally geodesic or of
positive constant curvature. This proves part (i) of Theorem 4.
Now, suppose that dimN > 2 and the mean curvature vector is not parallel.
From part (i), N is of positive constant curvature c. Since N is a totally umbilical
submanifold of constant mean curvature, (3.2) gives
(5.1)
R(X,Y;
Z,H) =
Q
for vector fields X, J^Z tangent to N. By using VR = 0. (1.1), (1.2), (2.5) and (5.1) we

[7] Totally umbilical submanifolds
in
symmetric spaces
135
find
(5.2)
a2
R{X,
Y; Z,
U)
=
g{U,
X)R(H, Y; Z,H)-g(U, Y)R(H, X; Z, H)
+ g(Y,Z)g(DxH,DuH)-g(X,Z)g(DYH,DuH),
for
U
tangent
to
JV.
Let X = U, Y= Z be
orthonormal vectors tangent
to
JV.
Then
(2.4)
and
(5.2) gives
(5.3) oc2k(H
A y) =
-|DxH|2
+
a2c-<x4,
where
c is the
constant sectional curvature
of
JV.
Since this is true for all orthonormal vectors
X and Y
tangent
to
JV
and dim
JV
> 2,
Dx
H
j
is
independent
of
A'.
In
particular, since
H
is not parallel,
|
Dx H
\
is
nonzero
for
any
unit vector A" tangent
to
JV
at
some point
p in N.
Let
U = X = £, and
summing
on i in (5.2) we
find
(5.4)
(n-l)R(H,Y\ Z,H) = a2
S(Y,
Z)-(n-
I)
<x4
g(Y,
Z)
+ g(DYH,DzH)-g(Y,Z)\DH\2,
by virtue of
(2.4),
where
j
D//|2
=
"£"=!
g(DE:
H,
£>£
H)
and S is
the Ricci tensor
of
JV.
Substituting
(5.4)
into
(5.2) we
obtain
(5.5) (/i-l)a2R(A\
Y\ Z.U)
= DH\2{cHY,U)g(X,Z)-g(X,U)g(Y,Z)}
+ cj(U,X){a2S(Y,Z)
+
g(DYH,DzH)}
-
g(U,
Y) {a2 S(X,
Z) +
g(Dx H,
Dz
H)}
+ (n-\){g(Y,Z)g(DxH,DL,H)-g(X,Z)g(DYH,DvH)}.
By setting
Y= Z = £, and
summing
on i, we get
(5.6)
(n-2)g(DxH,DLll)
=
a2S(A",
U)
+
r^g(X,
U)\DH\2--oc2rg(X,
U),
where
x =
Y."=i S(£;,£,•)
is the
scalar curvature
of
JV.
Since JV
is of
constant curvature,
(5.6)
shows that
(5.7)
g(DxH,DvH) = 0
for orthonormal vectors
A"
and U.
Because
Dx H is
nonzero
at
p
for all
unit vector
XeTpN.
(5.7) tells
us
that
H,DElH,...,DErH
are mutually orthogonal and they span
an (n+
1
(-dimensional linear subspace
of
the
normal space
Tp N.
This proves part
(ii) of
Theorem
4.

136 Bang-Yen Chen [8]
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B.
Y. Chen (1979), 'Extrinsic spheres in Riemannian manifolds', Houston J. Math. 5, 319 324.
B.
Y. Chen and T. Nagano (1977), Totally geodesic submanifolds of symmetric spaces. 1', Duke Math. J.
44,
745-755.
B.
Y. Chen andT. Nagano (1978),'Totally geodesic submanifolds of symmetric spaces. II', Duke Math. J.
45,
405-425.
S. I. Goldberg (1969), 'On conformally flat spaces with definite Ricci cun at
ure'.
Kodai Math. Sem. Rep.
21.
226-232.
S. Helgason (1968), Differential geometry, Lie groups and symmetric spates (Academic Press, New York).
T. Miyazawa and G. Chuman (1972), On certain subspaces of Riemannian recurrent spaces'. Tensor 23.
253-260.
J. A. Wolf (1963), 'Elliptic spaces in Grassmann manifolds'. Illinois J. Math. 7, 447-462.
Department of Mathematics
Michigan State University
East Lansing, Michigan 48824
U.S.A.