On spacelike hypersurfaces of constant sectional curvature lorentz manifolds

Let x:Mn→M¯n+1 be an n  -dimensional spacelike hypersurface of a constant sectional curvature Lorentz manifold M¯. Based on previous work of S. Montiel, L. Alías, A. Brasil and G. Colares studied what can be said about the geometry of M   when M¯ is a conformally stationary spacetime, with timelike conformal vector field K  . For example, if MnMn has constant higher order mean curvatures HrHr and Hr+1Hr+1, they concluded that MnMn is totally umbilical, provided Hr+1≠0Hr+1≠0 on it. If div(KK) does not vanish on MnMn they also proved that MnMn is totally umbilical, provided it has, a priori, just one constant higher order mean curvature.In this paper, we compute Lr(Sr)Lr(Sr) for such an immersion, and use the resulting formula to study both r  -maximal spacelike hypersurfaces of M¯, as well as, in the presence of a constant higher order mean curvature, constraints on the sectional curvature of M that also suffice to guarantee the umbilicity of M  . Here, by LrLr we mean the linearization of the second order differential operator associated to the r  -th elementary symmetric function SrSr on the eigenvalues of the second fundamental form of x.