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We address the problem of determining the hypersurfaces $fcolon M^{n} to mathbb{Q}_s^{n+1}(c)$ with dimension $ngeq 3$ of a pseudo-Riemannian space form of dimension $n+1$, constant curvature $c$ and index $sin {0, 1}$ for which there exists another isometric immersion $tilde{f}colon M^{n} to mathbb{Q}^{n+1}_{tilde s}(tilde{c})$ with $tilde{c} eq c$. For $ngeq 4$, 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 PDEs. 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.
We study conformally flat hypersurfaces $fcolon M^{3} to Q^{4}(c)$ with three distinct principal curvatures and constant mean curvature $H$ in a space form with constant sectional curvature $c$. First we extend a theorem due to Defever when $c=0$ and
We revisit Allendoerfer-Weils formula for the Euler characteristic of embedded hypersurfaces in constant sectional curvature manifolds, first taking some time to re-prove it while demonstrating techniques of [2] and then applying it to gain new understanding of isoparametric hypersurfaces.
In this paper we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson-Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen hypersurfaces in Loren
In this paper we give a complete local parametric classification of the hypersurfaces with dimension at least three of a space form that carry a totally geodesic foliation of codimension one. A classification under the assumption that the leaves of t
The Alexandrov Soap Bubble Theorem asserts that the distance spheres are the only embedded closed connected hypersurfaces in space forms having constant mean curvature. The theorem can be extended to more general functions of the principal curvatures