Characterization of Compact Riemannian and Spacelike Hypersurfaces in Einstein Manifolds
Mohammed GuediriWe explore rigidity results for compact Riemannian and spacelike hypersurfaces in Einstein manifolds. We show that for a compact Riemannian (or spacelike) hypersurface (M,g) in an Einstein Riemannian (or Lorentzian) manifold (M¯,g¯) which admits a closed conformal vector field ξ¯ (assumed to be timelike in the Lorentzian case), the hypersurface (M,g) is an extrinsic sphere in (M¯,g¯) if and only if it is not totally geodesic and the integral of the Ricci curvature of the hypersurface, taken in the direction of the tangential component ξ⊤ of the restriction ξ of ξ¯ to M, satisfies a certain lower bound. As an application of this result, we investigate compact spacelike hypersurfaces in generalized Robertson–Walker (GRW) spacetimes. Additionally, other related results are also provided.