A fundamental theorem for submanifolds in semi-Riemannian warped products

Abstract

In this paper we find necessary and sufficient conditions for a nondegenerate arbitrary signature manifold M n to be realized as a submanifold in the large class of warped product manifolds εI × a M N λ (c), where ε = ±1, a : I ⊂ R → R + is the scale factor and M N λ (c) is the N-dimensional semi-Riemannian space form of index λ and constant curvature c ∈ {−1, 1}. We prove that if M n satisfies Gauss, Codazzi and Ricci equations for a submanifold in εI × a M N λ (c), along with some additional conditions, then M n can be isometrically immersed into εI × a M N λ (c). This comprises the case of hypersurfaces immersed in semi-Riemannian warped products proved by M.A. Lawn and M. Ortega (see [6]), which is an extension of the isometric immersion result obtained by J. Roth in the Lorentzian products S n × R 1 and H n × R 1 (see [12]), where S n and H n stand for the sphere and hyperbolic space of dimension n, respectively. This last result, in turn, is an expansion to pseudo-Riemannian manifolds of the isometric immersion result proved by B. Daniel in S n × R and H n × R (see [2]), one of the first generalizations of the classical theorem for submanifolds in space forms (see [13]). Although additional conditions to Gauss, Codazzi and Ricci equations are not necessary in the classical theorem for submanifolds in space forms, they appear in all other cases cited above.