A Fundamental Theorem for Hypersurfaces in Semi-Riemannian Warped Products

Abstract

We give necessary and sufficient conditions for a semi-Riemannian manifold of arbitrary signature to be locally isometrically immersed into certain warped products. Then, we describe a way to use the structure equations of such immersions to construct foliations of marginally trapped surfaces in a four-dimensional Lorentzian spacetimes. We point out that, sometimes, Gauss and Codazzi equations are not sufficient to ensure the existence of a local isometric immersion of a semi-Riemannian manifold as a hypersurface of another manifold. We finally give two low-dimensional examples to illustrate our results.

26 pages

Cited by in corpus (4)

• Complex and Lagrangian surfaces of the complex projective plane via Kählerian Killing Spin${}^c$ spinors
• A Fundamental Theorem for submanifolds of multiproducts of real space forms
• On Einstein hypersurfaces of $I{\times }_f{\mathbb{Q}}^n(c)$
• Isometric immersions into manifolds with metallic structures