A New Characterization of Totally Umbilical Hypersurfaces in de Sitter Space

It is shown that a compact spacelike hypersurface which is contained in the chronological future (or past) of an equator of de Sitter space is a totally umbilical round sphere if the kth mean curvature function Hk is a linear combination of Hk+1,∙∙∙, Hn. This is a new angle to characterize round spheres.


Introduction
Let 2 1 n R + be the (n + 2)-dimensional Lorentz-Minkowski space, namely, the real vector space 2 n R + endowed with the Lorentzian inner product , given by ( ) ( ) .
Then the n-dimensional de Sitter space is defined by It is well known that, for 2 n ≥ the de Sitter space : M is said to be a spacelike hypersurface if the induced metric via ψ is a Riemannian metric on n M , 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 1 1 n S + 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 H 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 ( ) H 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 M 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 M is a totally umbilical round sphere.In [6] the authors considered that k H is the linear combination of 1 , and proved: be a compact spacelike hypersurface in de Sitter space which is contained in the chronological future (or past) of an equator of M 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

Preliminaries
Throughout this paper we will deal with compact spacelike hypersurfaces in de Sitter space.Recall that every compact spacelike hypersurfaces n M in 1 1 n S + is diffeomorphic to an n-sphere [4] and, in particular, it is orientable.Then, there exists a timelike unit normal field N globally defined on n M .We will refer to N as the Gauss map of the immersion and we will say that n M is oriented by N .We will denote by ( ) ( ) A X dN X = − Associated to the shape operator of n M there are n algebraic invariants, which are the elementary symmetric functions k σ of its principal curvatures 1 , , , given by ( ) The kth mean curvature k H of the spacelike hypersurfaces is then defined by ( ) ( ) ( ) is the mean curvature of n M .On the other hand, when , k n = ( ) ( ) defines the Gauss-Kronecker curvature of the spacelike hypersurface, and for 2, (for details see [4]).The proof of our theorem makes an essential use of the following integral formulas for compact spacelike hypersurfaces in 1 1 n S + , which is developed in [4].Lemma 5 (Minkowski formulas) Let where dV is the n-dimensional volume element of n M with respect to the induced metric and the chosen orientation.

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 (for the details see the proof of Theorem 7 in [4]).On the other hand, since M 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 j H is positive for 1, , j n =  and 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 on M , with equality if and only if 1 which implies that M is an totally umbilical round sphere.It remains to prove (3).Using the assumption of theorem 4, that is and (2), we have Now we prove the second statement.Using and the Minkowski formulae, we have which implies that M 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.

1 1 nS
+ is the standard simply connected Lorentzian space form of positive constant sectional curvature.
spacelike hypersurface immersed into de Sitter space and let 2 1 n a R + ∈ a fixed arbitrary vector.For each 0, , 1 r n = −  the following formula holds:

M
of the past is similar).That means that by the Gauss map N which is in the same time-orientation as a , so that , that n H does not vanish on n