Chen’s Ricci inequalities and topological obstructions on null hypersurfaces of a Lorentzian manifold

Abstract

Abstract Given a null hypersurface of a Lorentzian manifold, we isometrically immerse a null hypersurface equipped with the Riemannian metric (induced on it by the rigging) into a Riemannian manifold suitably constructed on the Lorentzian manifold. We study the intrinsic and extrinsic geometry of such an isometric immersion and we link them to the null geometry of the null hypersurface in the Lorentzian manifold. In the course of this immersion, we find the basic relationships between the main extrinsic invariants and the main intrinsic invariants, named Chen-Ricci inequalities of the null hypersurface in the Lorentzian manifold. The findings prove a topological implication of these relationships.

Keywords

• Null hypersurface
• Rigging
• Closed normalization
• Associated Riemannian metric
• Ricci inequalities
• Minimal hypersurface