A New Characterization of Totally Umbilical Hypersurfaces in de Sitter Space ()
Then the n-dimensional de Sitter space is defined by
. It is well known that, for 
the de Sitter space 
is the standard simply connected Lorentzian space form of positive constant sectional curvature. A smooth immersion 
of an 
-dimensional connected manifold 
is said to be a spacelike hypersurface if the induced metric via 
is a Riemannian metric on
, which, as usual, is also denoted by
.
The interest for the study of spacelike hypersurfaces in de Sitter space is motivated by the fact that such hypersurfaces exhibit nice Bernstein-type properties. In 1977, Goddard [1] conjectured that the only complete spacelike hypersurfaces with constant mean curvature in 
should be the totally umbilical ones. This conjecture motivated the work of an important number of authors who considered the problem of characterizing the totally umbilical spacelike hypersurfaces of de Sitter space. In [2], Montiel showed that the only compact spacelike hypersurfaces in 
with constant mean curvature 
were the totally umbilical round spheres. More recently, Cheng and Ishikawa [3] have shown that the totally umbilical round spheres are the only compact spacelike hypersurfaces in de Sitter space with constant scalar curvature
.
The natural generalization of mean and scalar curvature for a spacelike hypersurface in de Sitter space are the kth mean curvature 
for
. Actually, 
is the mean curvature and 
is, up to a constant, the scalar curvature of the hypersurface. In [4], Aledo, jointly with Alias and Romero, developed some integral formulas for compact spacelike hypersurfaces in 
and applied them in order to characterize the totally umbilical round spheres of
.
Theorem 1([4], Theorem 7) Let 
be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of
. If 
is constant for some
, then 
is a totally umbilical round sphere.
Since 
by definition, the result above can be read as follows: if 
is constant for some
, then 
is a totally umbilical round sphere. In [5], Alias extended Theorem 1 in the following way.
Theorem 2 ([5]) Let 
be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of
. If 
does not vanish on 
and the ratio 
is constant for some
, then 
is a totally umbilical round sphere.
In [6] the authors considered that 
is the linear combination of
, and proved:
Theorem 3 ([6]) Let 
be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of
. If there are nonnegative constants
, at least one 
is positive, such that 
holds on
, then 
is a totally umbilical round sphere.
In this paper, we will show another characterization of totally umbilical round sphere, which extends Theorems 1 and 2 above.
Theorem 4 Let 
be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of
. If 
does not vanish on
for some fixed
, 
, there exist constants 
such that
on
, then 
is a totally umbilical round sphere.
there are 
constants 
such that 
on
, then 
is a totally umbilical round sphere.
Remark.
Note in some special cases the condition 
does not vanish should can be dropped, for examples, only one coefficient 
case. However in general cases we can not drop it now.
The corresponding theorem characterizes ellipsoids also holds in affine differential geometry.
2. Preliminaries
Throughout this paper we will deal with compact spacelike hypersurfaces in de Sitter space. Recall that every compact spacelike hypersurfaces 
in 
is diffeomorphic to an n-sphere [4] and, in particular, it is orientable. Then, there exists a timelike unit normal field 
globally defined on
. We will refer to 
as the Gauss map of the immersion and we will say that 
is oriented by
.
We will denote by 
the shape operator of 
in 
with respect to
, which is given by
Associated to the shape operator of 
there are 
algebraic invariants, which are the elementary symmetric functions 
of its principal curvatures 
given by
The kth mean curvature 
of the spacelike hypersurfaces is then defined by
When 
is the mean curvature of
. On the other hand, when 
defines the Gauss-Kronecker curvature of the spacelike hypersurface, and for 
is, up to a constant, the scalar curvature 
of
, since 
(for details see [4]).
The proof of our theorem makes an essential use of the following integral formulas for compact spacelike hypersurfaces in
, which is developed in [4].
Lemma 5 (Minkowski formulas) Let 
be a compact spacelike hypersurface immersed into de Sitter space and let 
a fixed arbitrary vector. For each 
the following formula holds:
where 
is the n-dimensional volume element of 
with respect to the induced metric and the chosen orientation.
3. Proof of the Theorem 4
Let us assume, for instance, that the hypersurface 
is contained in the future of the equator determined by a unit timelike vector 
(the case of the past is similar). That means that
Let us orient 
by the Gauss map 
which is in the same time-orientation as
, so that 
Since the height function 
is negative on
, by compactness there exists a point 
where it attains its maximum
Therefore, its gradient vanishes at that point, 
, and its Hessian satisfies
for all 
(for the details see the proof of Theorem 7 in [4]). On the other hand, since
and
then
Therefore, choosing 
a basis of principal directions at the point 
we conclude that
(1)
for each 
In particular, 
are positive. The mean curvature functions 
is positive on 
(recall that 
does not vanish on 
by assumption). Therefore, from the proof of Lemma 1 in [7] and taking into account the sign convention in our definition of the higher order mean curvature, it follows that every 
is positive for 
and
(2)
with equality at any stage only at umbilical points.
Let us start proving the first statement of Theorem 4. Using
and the Minkowski formulae, we have
That is,
Now we claim that
(3)
on
, with equality if and only if
. Assume that (3) is true. Then, since 
on
, we conclude that
which implies that 
is an totally umbilical round sphere.
It remains to prove (3). Using the assumption of theorem 4, that is 
and (2), we have
(4)
Now we prove the second statement. Using
and the Minkowski formulae, we have
That is,
Now we claim that
(5)
on
, with equality if and only if
. Assume that (5) is true. Then, since 
on
, we conclude that
which implies that 
is an totally umbilical round sphere. It remains to prove (5). As in the first proof, using the assumption of theorem 4, that is
, and (2) we known
This completes the proof of the Theorem 4.
Funding
This work is supported by grant (No.U1304101 and 11171091) of NSFC and NSF of Henan Province (No.132300410141).
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