Hypersurfaces of two space forms and conformally flat hypersurfaces

Abstract

We address the problem of determining the hypersurfaces $f\colon M^{n} \to \mathbb{Q}_s^{n+1}(c)$ with dimension $n\geq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $s\in \{0, 1\}$ for which there exists another isometric immersion $\tilde{f}\colon M^{n} \to \mathbb{Q}^{n+1}_{\tilde s}(\tilde{c})$ with $\tilde{c}\neq c$. For $n\geq 4$, we provide a complete solution by extending results for $s=0=\tilde s$ by do Carmo and Dajczer and by Dajczer and the second author. Our main results are for the most interesting case $n=3$, and these are new even in the Riemannian case $s=0=\tilde s$. In particular, we characterize the solutions that have dimension $n=3$ and three distinct principal curvatures. We show that these are closely related to conformally flat hypersurfaces of $\mathbb{Q}_s^{4}(c)$ with three distinct principal curvatures, and we obtain a similar characterization of the latter that improves a theorem by Hertrich-Jeromin. We also derive a Ribaucour transformation for both classes of hypersurfaces, which gives a process to produce a family of new elements of those classes, starting from a given one, in terms of solutions of a linear system of PDE's. This enables us to construct explicit examples of three-dimensional solutions of the problem, as well as new explicit examples of three-dimensional conformally flat hypersurfaces that have three distinct principal curvatures.