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 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 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 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.

Superfícies do tipo Enneper

“…Inspired by the recent results obtained in (CHION; TOJEIRO, 2021) and (TASSI; TO-JEIRO, In preparation.) we give an explicit description of surfaces of Enneper type for which the lines of curvature of one family are contained either in concentric spheres, parallel planes, or planes that intersect along a common straight line.…”

“…The Wente tori are surfaces of Enneper type, as shown in (ABRESCH, 1987) and (SPRUCK, 1988). Chion and Tojeiro (2021) in a recent work introduced the notion of Ribaucour partial tubes and generalized the results described in Bianchis's book for the corresponding hypersurfaces of Enneper type in R n+1 . The authors obtained in particular a new description of the special class of surfaces of Enneper type with the property that the spheres that contain the lines of curvature correspondent to one of its principal curvatures are all centered on a common straight Introduction line, called Joachimsthal surfaces, based on the conformal diffeomorphism of H 2 × R onto R 3 ∖ R.…”

“…The above complete description of all Joachimsthal surfaces in R 3 in terms of the conformal covering map Φ : H 2 × R → R 3 ∖ R was first obtained by Chion and Tojeiro (2021), and yields a new description for such surfaces. Furthermore, Corollary 3.1.2.1 allows us to recover the classical fact that, for surfaces free of umbilical points, the lines of curvature in one family lie in planes passing through a common line r if and only if the lines of curvature in the second family lie on spheres whose centers are on r and which cut the surface orthogonally, see (EISENHART, 1909, p. 308) and(BIANCHI, 1922, §193).…”