Euclidean hypersurfaces with a totally geodesic foliation of codimension one

TL;DR

This paper classifies Euclidean hypersurfaces with a specific geometric property, showing that certain rotation hypersurfaces are characterized by their warped product structure, and also discusses the local version of the problem.

Contribution

It provides a complete classification of hypersurfaces with a totally geodesic foliation of codimension one in Euclidean space, including a characterization of rotation hypersurfaces.

Findings

Rotation hypersurfaces with complete profiles are characterized by warped product structure.

The classification includes both global and local cases.

The results deepen understanding of hypersurfaces with special foliations.

Abstract

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.