On spacelike hypersurfaces with two distinct principal curvatures in Lorentzian space forms

Abstract

We investigate the spacelike hypersurfaces in Lorentzian space forms N n+1 1 (c) (n 4) with two distinct non-simple principal curvatures without the assumption that the (high order) mean curvature is constant. We prove that any spacelike hypersurface in Lorentzian space forms with two distinct non-simple principal curvatures is locally conformal to the Riemannian product of two constant curved manifolds. We also obtain some characterizations for hyperbolic cylinders in Lorentzian space forms in terms of the trace free part of the second fundamental form.

On hypersurfaces with two distinct principal curvatures in space forms

“…It should be noted that this lemma has been verified in [7], here we include the proof for readers' convenience. We shall only prove (3), and (1) and (2) can be proved similarly.…”

“…(2) The basic idea of the present paper can be used to study space-like hypersurfaces in Lorentzian space forms and we can obtain the Lorentzian versions of the main results of this paper (see [7]). …”