Spacelike Hypersurfaces in de Sitter Space

A closed conformal vector field in de Sitter space S1n+1c¯ induces a vector field on a spacelike hypersurface M of S1n+1c¯, referred to as the induced vector field on M. This article investigates the characterization of compact spacelike hypersurfaces in de Sitter space without assuming the constancy of the mean curvature. Specifically, we establish that under certain conditions, a compact spacelike hypersurface in S1n+1c¯ is a sphere, that is, a totally umbilical hypersurface with constant mean curvature. We also present a different characterization of compact spacelike hypersurfaces in de Sitter space as spheres by using a lower bound on the integral of the Ricci curvature of the compact hypersurface in the direction of the induced vector field. We also consider de Sitter space as a Robertson–Walker space and provide several characterizations of spheres within its spacelike hypersurfaces.