A fundamental theorem for submanifolds in semi-Riemannian warped products

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}$^{N}{\lambda}©$,where$\varepsilon=\pm 1,
a:I\subset$\mathbb{R}$\to$\mathbb{R}$^+$isthescalefactorand$$\mathbb{M}$^{N}{\lambda}©$isthe$N$-dimensional semi-Riemannian space form of index $\lambda$ and constant curvature $c\in \{-1,1\}\mathrm{.}$ We prove that if $M^n$ satisfies Gauss, Codazzi and Ricci equations for a submanifold in $\epsilon I{\times }_a{\mathbb{M}}_{\lambda }^N(c)$, along with some additional conditions, then $M^n$ can be isometrically immersed into $\varepsilon I\times_a$\mathbb{M}$^{N}_{\lambda}©$. 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.