Hypersurfaces of space forms carrying a totally geodesic foliation

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 already given in [6] 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.

Ribaucour partial tubes and hypersurfaces of Enneper type

“…In [5] and [6], the authors studied the problem of determining the hypersurfaces of dimension at least three of a space form that carry a totally geodesic foliation of codimension one. The initial motivation of this work was to investigate the similar problem that one can pose by assuming the foliation to be spherical instead of being totally geodesic.…”

“…In the solution of the problem addressed in [5] and [6], one main example of a hypersurface of a space form that carries a totally geodesic foliation of codimension one is a partial tube over a smooth regular curve. Partial tubes are submanifolds that are generated by starting with any submanifold whose normal bundle has a parallel and flat subbundle, taking a submanifold in the fiber of that subbundle at a given point, and then parallel transporting it along the former submanifold with respect to its normal connection.…”