Hypersurfaces of two space forms and conformally flat hypersurfaces

Abstract

We address the problem of determining the hypersurfaces \(f:M^{n} \rightarrow \mathbb {Q}_s^{n+1}(c)\) with dimension \(n\ge 3\) 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 \(\tilde{f}:M^{n} \rightarrow \mathbb {Q}^{n+1}_{\tilde{s}}(\tilde{c})\) with \(\tilde{c}\ne c\). For \(n\ge 4\), we provide a complete solution by extending results for \(s=0=\tilde{s}\) by do Carmo and Dajczer (Proc Am Math Soc 86:115–119, 1982) and by Dajczer and Tojeiro (J Differ Geom 36:1–18, 1992). 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 (Beitr Algebra Geom 35:315–331, 1994).