On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds

Let x : M-n -> (M) over bar (n+1) be an n-dimensional spacelike hypersurface of a constant sectional curvature Lorentz manifold (M) over bar. Based on previous work of S. Montiel, L. Alias, A. Brasil and G. Colares studied what can be said about the geometry of M when (M) over bar is a conformally stationary spacetime, with timelike conformal vector field K. For example, if M-n has constant higher order mean curvatures H-r and Hr+1, they concluded that M-n is totally umbilical, provided Hr+1 not equal 0 on it. If div(K) does not vanish on M-n they also proved that M-n is totally umbilical, provided it has, a priori, just one constant higher order mean curvature. In this paper, we compute L-r(S-r) for such an immersion, and use the resulting formula to study both r-maximal spacelike hypersurfaces of (M) over bar, 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 L-r we mean the linearization of the second order differential operator associated to the r-th elementary symmetric function S, on the eigenvalues of the second fundamental form of x. (c) 2005 Elsevier B.V. All rights reserved.