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 $\varepsilon I\times_a\mathbb{M}^{N}_{\lambda}(c)$, where $\varepsilon=\pm 1,\ a:I\subset\mathbb{R}\to\mathbb{R}^+$ is the scale factor and $\mathbb{M}^{N}_{\lambda}(c)$ is the $N$-dimensional semi-Riemannian space form of index $\lambda$ and constant curvature $c\in\{-1,1\}.$ We prove that if $M^n$ satisfies Gauss, Codazzi and Ricci equations for a submanifold in $\varepsilon I\times_a\mathbb{M}^{N}_{\lambda}(c)$, along with some additional conditions, then $M^n$ can be isometrically immersed into $\varepsilon I\times_a\mathbb{M}^{N}_{\lambda}(c)$. 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 $\mathbb{S}^n\times\mathbb{R}_1$ and $\mathbb{H}^n\times\mathbb{R}_1$ (see [12]), where $\mathbb{S}^n$ and $\mathbb{H}^n$ 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 $\mathbb{S}^n\times\mathbb{R}$ and $\mathbb{H}^n\times\mathbb{R}$ (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.