Rotation hypersurfaces in semi-Riemannian space forms of arbitrary index

Abstract

Let n, ν be integers such that n≥2 and 0≤ν≤n+1. Let Mc,νn+1 be a (n+1)-dimensional semi-Riemannian space form of constant sectional curvature c and index ν. We give the parametrizations of rotation hypersurfaces in Mc,νn+1. Adding the condition for the r-th mean curvature to be constant, we show the study of these hypersurfaces is reduced to the study of a function on two variables. As an illustrative example, we study the particular case of rotation hypersurfaces with constant scalar curvature in Mc,νn+1, determining the values of n,c,ν for which these hypersurfaces are isoparametric. In the final part we study the variation of the second fundamental form of rotation hypersurfaces in order to obtain some rigidity results.