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Register a new userHypersurfaces of space forms carrying a totally geodesic foliation
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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 the foliation are complete was given in cite{drt} for Euclidean hypersurfaces. We prove that there exists exactly one further class of local examples in Euclidean space, all of which have rank two. We also extend the classification under the global assumption of completeness of the leaves for hypersurfaces of the sphere and show that there exist plenty of examples in hyperbolic space.
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We classify the hypersurfaces of Euclidean space that carry a totally geodesic foliation with complete leaves of codimension one. In particular, we show that rotation hypersurfaces with complete profiles of codimension one are characterized by their warped product structure. The local version of the problem is also considered.
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.
In the present paper we study geometric structures associated with webs of hypersurfaces. We prove that with any geodesic (n+2)-web on an n-dimensional manifold there is naturally associated a unique projective structure and, provided that one of web foliations is pointed, there is also associated a unique affine structure. The projective structure can be chosen by the claim that the leaves of all web foliations are totally geodesic, and the affine structure by an additional claim that one of web functions is affine. These structures allow us to determine differential invariants of geodesic webs and give geometrically clear answers to some classical problems of the web theory such as the web linearization and the Gronwall theorem.
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 Lorentzian space forms and provide a local characterization of such hypersurfaces.
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