Hypersurfaces of space forms carrying a totally geodesic foliation

TL;DR

This paper provides a comprehensive local classification of hypersurfaces in space forms with a codimension-one totally geodesic foliation, extending previous results and identifying new examples especially in Euclidean and hyperbolic spaces.

Contribution

It offers a complete local classification of such hypersurfaces, including new classes in Euclidean space and examples in hyperbolic space, under various completeness assumptions.

Findings

Classified hypersurfaces with totally geodesic foliations in space forms.

Identified a unique class of Euclidean hypersurfaces with rank two.

Constructed numerous examples in hyperbolic space.

Abstract

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.