533
Internat. J. Math. & Math. Sci.
VOL. 14 NO. 3 (1991) 533-536
SUBMANIFOLDS OF EUCLIDEAN SPACE WITH PARALLEL
MEAN CURVATURE VECTOR
TAHSIN GHAZAL and SHARIEF DESHMUKH
Department of Mathematics
College of Science
P.O. Box 2455
King Saud University
Riyadh 11451, Saudi Arabia
(Received November 21, 1989 and in revised form October 19, 1990)
The object of the paper is to study some compact
ABSTRACT.
submaniforlds
vector
in
the
Euclidean space Rn
First we prove
parallel in the normal bundle.
is
does not exist an n-dimensional compact
there
totally
real
parallel.
in
Positively
curvature
R 2n
curved
vector
particular
of
constant
curvature
Finally
submanifolds in R n
are
are
flat.
homology spheres.
we
show
that
parallel
with
The
last
is
totally
mean
parallel
and
compact
mean
result
for even dimensional submanifolds implies that
poincare’ characteristic
Euler
connected
simply
we show that the n-dimensional compact
Then
curvature
that
submanifold in R 2n whose mean curvature vector
submanifolds
real
mean curvature
whose
in
their
class is positive, which for the
class of compact positively curved submanifolds admiting isometric
immersion with parallel mean curvature vector in R
answers the
n,
problem of Chern and Hopf.
KEY WORDS AND PHRASES.
Submanifolds, totally real submanifolds,
homology sphere, Euler-poincare’ characteristic.
1980 AMS SUBJECT CLASSIFICATION:
i.
Let
Riemannian
bundle
9,
g
be
53C21, 53C25, 53C40.
connection.
n,
corresponding
n
If M is a submanifold of R with normal
the flat metric of R
be
the
then the connection V induces the Riemannian connection
534
T. GHAZAL AND S. DESHMUKH
#
V on M and the connection
in the normal bundle u, and we have
VxY=VxY+h(X,Y), VxN=-ANX+VxN,
where
X,Y c(M), N u
the second fundmental form h(X,Y)
(i.i)
is related
ANX
to
by
g (h (X, Y) N)
g(ANX,Y
fields
The mean curvature vector H of M is given by H
on M.
i=l
I/n
VxH
if
If H
X(M).
0
to be a minimal submanifold.
submanifold of R
structure
that is,
to
said
bundle
n,
even
The
orthonormal
The mean curvature vector H is said to be
0,
vector
of
e n) is a local
where {e 1,e 2,
h(ei,ei),
frame of M.
is the Lie-algebra
and I(M)
parallel
at each point of M, then M is said
It is known that if M is a compact
then M is not a minmal submanifold (cf. [i]).
dimensional
Euclidean
space
R 2n
complex
has
J which is parallel with respect to the connection V
R 2n is a kaehler manifold.
A submanifold M of R 2n is
totally real if JTM
be
of
In
M.
=-u,
where TM is
n and
the case dim M
M
is
tangent
the
totally
real
submanifold of R 2n, using (i.I), we obtain
VxJY
In
JVxY
this
and h(X,Y)
(1.2)
JAjyX, X,YI(M).
section we study the
n-dimensional
totally
real
submanifold M of R 2n with parallel mean curvature vector H,
first
2.
under the topological restriction on M that,
it is compact and its
first Betti number is zero, and then under the geometric restrict-
ion that it is a space of constant curvature.
There does not exist an n-dimensional compact
THEOREM 2.1.
totally
to
zero
totally
real
real submanifold with first Betti number equal
and with parallel mean curvature vector in R 2n.
Let
PROOF.
M
an n-dimensional compact
be
submanifold of R 2n with parallel mean curvature vector
H.
Then
JH is a parallel vector field on M.
The 1-form D dual to JH is also parallel and hence harmonic.
If
HI(M;
mean
)
that
0,
then
n
But this would
and hence H must vanish.
M is a compact minimal submanifold of R 2n,
which
is
impossible (cf. [i]).
THEOREM
totally
real
2.2.
Let
submanifold
M be an n-dimensional
of constant
parallel mean curvature vector.
curvature
Then M is flat.
2)
n
in
compact
R 2n
with
SUBMANIFOLDS OF EUCLIDEAN SPACE
PROOF.
is
then M
curvature is nonzero constant,
the
If
535
irreducible and cannot admit a nonzero parallel vector field JH.
In
3.
this section we shall be concerned with
with parallel mean curvature vector in
submanifolds
curved
positively
the
n.
R
We prove the following.
THEOREM
with parallel mean curvature vector
submanifold
curved
positively
Let M be a compact and connected
3.1.
n.
in
R
0,
the
Then M is a homology sphere.
I
M is compact,
Sicne
PROOF.
function
N=I
of M as submanifold of R
n,
M
If
R n is the immersion
then the height function
M of
fN:
R is
M
The hessian of the height function
g((p),N).
fN(p)
at a critical point p
at p.
H.
VxH
Define the unit normal
is a non-zero constant.
[]
vector field N on M by
defined by
connected and
fN
is given by the weingarten map A N
The curvature tensor R of M is given by
R(X,Y;Z,W)
g(h(Y,Z),h(X,W))-g(h(X,Z),h(Y,W)),
from which the Ricci tensor Ric of M is obtained as
Ric(X,Y)
where
is
the
ng(h(X,Y),H)-
(el,e2,...,en}
=ig(h(X,ei),h(Y,ei)),
is a local orthonormal frame of M.
positively curved and for a unit vector field X,
sum of the sectional curvatures,
(3.1) we obtain g(h(X,X),H) > 0.
for
each unit vector field X
eigenvalues
of
Ric(X,X) >
Sicne M
Ric(X,X) is
from
Thus
0.
This given that
.
g(ANX,X
>
0,
This proves that all the
(M).
AN are positive at each point of
height function
fN
i=l,2,...,n-l.
Using Morse inequalities we obtain
H I(M,R)
(3.1)
Thus
the
has no non-degenerate critical points of index
H n-l(s,R)
0,.
0.
Since M is compact, we get that M is a homology sphere.
COROLLARY
3.1.
The
real
RPm
projective space
and
the
complex projective space CP 2 cannot be isometrically immersed in
Rn with parallel mean curvature vector.
Combining Theorem 2.1 with Theorem 3.1, we get
COROLLARY
3.2.
There
does
not exist
an
n-dimenstional
compact and connected positively curved totally real
in R 2n with parallel mean curvature vector.
submanifold
536
T. GHAZAL AND S. DESHMUKH
Remark.
The Chern-Hopf
characteristic
manifold
positively
class
problem
is "The
of an even dimensional
M satisfies (M) > 0".
Eu]er-poincare’
positively
For class of even
curved
dimensional
compact and connected manifolds which admit
isometric commersion in R n with parallel mean curvature vector we
curved
have the following corollary to Theorem 3.1.
COROLLARY
3.3.
Let
M be an even dimensional compact
connected positively curved submanifold of R n with parallel
curvature vector. Then (M) > 0.
and
mean
ACKNOWLEDGEMENT
The research is supported by the grant No.
(Math/1409/05) of
the Research Center, College of Science, King Saud University
REFERENCES
i.
Kobayashi, S. and Nomizu, K., Foundations of differential
geometry, Vol II, Interscience tract, New York, (1069).
2.
Milnor, J., Morse Theory, Ann. of Math. Studies, Princeton
University Dress, Princeton, (1963).
3.
Weinstein, A., Positively Curved n-manifolds in Rn+2, J. Diff
Geom. 4 1-4 (1970)