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On spacelike hypersurfaces of constant sectional curvature Lorentz manifolds. (English) Zbl 1102.53044

The paper deals with n-dimensional space-like hypersurfaces of constant sectional curvature Lorentz manifolds given by an immersion of the form x:M n M ˜ n+1 · The author first computes L r (S r ) for such immersions where L r :𝒟(M)𝒟(M) is the second order differential operator defined as

L r (f)=tr(P r Hessf)

where P r is the r-th Newton transformation on M· Then the author makes use of the formula not only to consider r-maximal space-like hypersurfaces of M ˜ but also to obtain conditions for the sectional curvature of M to make it umbilic in the case of a constant higher order mean curvature. One of the main results of the paper in terms of applications of the formula for L r (S r ) is Theorem 1, i.e, if x:M n M ˜ c n+1 is a closed space-like hypersurface of a time-oriented Lorentz manifold of constant sectional curvature c0 and M is of constant scalar curvature R such that

(a) c(n-2 n)<Rc, then M is totally umbilical,

(b) c(n-2 n)Rc and S 3 0, then M is totally umbilical,

(c) c(n-2 n)R<c, then M has constant mean curvature.

Moreover, the case of c(n-2 n)<R<c is considered under some assumptions on the mean curvature H of M in Theorem 2.

53C42Immersions (differential geometry)
53C99Global differential geometry