In this paper we develop general Minkowski-type formulae for compact spacelike hypersurfaces immersed into conformally stationary spacetimes, that is, Lorentzian manifolds admitting a timelike conformal field. We apply them to the study of the umbilicity of compact spacelike hypersurfaces in terms of their r-mean curvatures. We derive several uniqueness results, for instance, compact spacelike hypersurfaces are umbilical if either some of their r-mean curvatures are linearly related or one of them is constant.

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Given a generalized M n+1 = I × φ F n Robertson-Walker spacetime we will classify strongly stable spacelike hypersurfaces with constant mean curvature whose warping function verifies a certain convexity condition. More precisely, we will show that given x : M n → M n+1 a closed spacelike hypersurfaces of M n+1 with constant mean curvature H and the warping function φ satisfying φ ′′ ≥ max{Hφ ′ , 0}, then M n is either minimal or a spacelike slice Mt 0 = {t 0 } × F , for some t 0 ∈ I.

Let φ : M → S n+1 ⊂ R n+2 be an immersion of a complete n-dimensional oriented manifold. For any v ∈ R n+2 , let us denote by v : M → R the function givenWe will prove that if M has constant mean curvature, and, for some v = 0 and some real number λ, we have that v = λf v , then, φ(M) is either a totally umbilical sphere or a Clifford hypersurface. As an application, we will use this result to prove that the weak stability index of any compact constant mean curvature hypersurface M n in S n+1 which is neither totally Dedicated to the memory of Professor Luis J. Alías-Pérez. 688 L.J. Alías et al.umbilical nor a Clifford hypersurface and has constant scalar curvature is greater than or equal to 2n + 4.

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