A Fundamental Theorem for Hypersurfaces in Semi-Riemannian Warped Products

Abstract

We give necessary and sufficient conditions for a semi-Riemannian manifold of arbi- trary signature to be locally isometrically immersed into a warped product ±I ×a M^n (c), where I ⊂ R and M^n (c) is a semi-Riemannian space of constant nonzero sectional cur- vature. 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 space- times. 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.