Imersões isométricas em produtos warped. - Ribeiro, Carlos Augusto David (2019)
In this work, 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 MNλ(c), where ε = ±1, a : I ⊂ R → R+ is the scale factor and MNλ(c) is the N-dimensional semi-Riemannian space form of index λ and constant curvature c ∈ {−1, 1}. We also discussed the necessary and sufficient conditions for a nondegenerate arbitrary signature manifold M n to be realized as a submanifold in the class of warped product manifolds MN1 k1(c 1 ) × h MN2 k2 (c 2 ). In the case of submanifolds of εI × a MNλ(c), we prove that if M n satisfies Gauss, Codazzi and Ricci equations for a submanifold in εI × a MNλ(c), along with some additional conditions, then M n can be isometrically immersed into εI × a MNλ(c). This comprises the case of hypersurfaces immersed in semi-Riemannian warped products proved by M.A. Lawn and M. Ortega (see Lawn and Ortega (2015)), which is an extension of the isometric immersion result obtained by J. Roth in the Lorentzian products Sn × R1 and Hn × R1 (see Roth (2011)), where Sn and Hn 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 Sn × R and Hn × R (see Daniel (2009)), one of the first generalizations of the classical theorem for submanifolds in space forms (see Tenenblat (1971)). In the case of submanifolds of warped products MN1 k1 (c 1 ) × h MN2 k2 (c 2 ), we discussed the necessary and sufficient conditions, as well as displaying a result for the case where the warping function has a conformal Hessian, which generalizes not only all the articles mentioned above, but also in part the result obtained in Lira, Tojeiro, and Vit´orio (2010).