Rigidity of extrinsic spheres in Einstein spacetimes and the Goddard conjecture

Goddard's conjecture asserts that within de Sitter space, a spacelike hypersurface that is complete and has constant mean curvature must exhibit total umbilicity. Although counterexamples show that the conjecture does not hold in full generality, it remains valid under additional geometric assumptions, notably in the compact case. In this paper, we establish a broad extension of Goddard's conjecture to compact, spacelike hypersurfaces in Lorentzian manifolds that admit a timelike conformal vector field. Under a natural integral condition involving the ambient Ricci tensor and a mild assumption of the behavior of the mean curvature along the induced tangential flow, we prove that the hypersurface must be totally umbilical. As an application of our approach, we establish rigidity phenomena in Einstein Lorentzian manifolds and retrieve, as a special instance, the well-known classification of compact, spacelike hypersurfaces with constant mean curvature in de Sitter space as round spheres.