Ribaucour partial tubes and hypersurfaces of Enneper type

Abstract

In this article we introduce the notion of a Ribaucour partial tube and use it to derive several applications. These are based on a characterization of Ribaucour partial tubes as the immersions of a product of two manifolds into a space form such that the distributions given by the tangent spaces of the factors are orthogonal to each other with respect to the induced metric, are invariant under all shape operators, and one of them is spherical. Our first application is a classification of all hypersurfaces with dimension at least three of a space form that carry a spherical foliation of codimension one, extending previous results by Dajczer, Rovenski and the second author for the totally geodesic case. We proceed to prove a general decomposition theorem for immersions of product manifolds, which extends several related results. Other main applications concern the class of hypersurfaces of $\mathbb{R}^{n+1}$ that are of Enneper type, that is, hypersurfaces that carry a family of lines of curvature, correspondent to a simple principal curvature, whose orthogonal $(n-1)$-dimensional distribution is integrable and whose leaves are contained in hyperspheres or affine hyperplanes of $\mathbb{R}^{n+1}$. We show how Ribaucour partial tubes in the sphere can be used to describe all $n$-dimensional hypersurfaces of Enneper type for which the leaves of the $(n-1)$-dimensional distribution are contained in affine hyperplanes of $\mathbb{R}^{n+1}$, and then show how a general hypersurface of Enneper type can be constructed in terms of a hypersurface in the latter class. We give an explicit description of some special hypersurfaces of Enneper type, among which are natural generalizations of the so called Joachimsthal surfaces.