A Fundamental Theorem for Hypersurfaces in Semi-Riemannian Warped Products
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Differential geometryGeneral relativity and quantum cosmologyMathematical physics
SponsorshipSpanish MEC-FEDER Grant MTM2007-60731 and Junta de Andalucía Grant P09-FQM-4496 (with FEDER funds.)
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.