A construction of Sbrana-Cartan hypersurfaces in the discrete class

TL;DR

This paper constructs numerous examples of Sbrana-Cartan hypersurfaces in the discrete class, advancing understanding of their existence and properties, and suggesting directions toward a full classification.

Contribution

It provides a broad construction of Sbrana-Cartan hypersurfaces, addressing the existence question and contributing to the classification problem.

Findings

Constructed many examples of Sbrana-Cartan hypersurfaces

Confirmed the existence of hypersurfaces of both hyperbolic and elliptic types

Indicated progress toward classifying these hypersurfaces

Abstract

The classical classifications of the locally isometrically deformable Euclidean hypersurfaces obtained by U. Sbrana in 1909 and E. Cartan in 1916 includes four classes, among them the one formed by submanifolds that allow just a single deformation. The question of whether these Sbrana-Cartan hypersurfaces do, in fact, exist was not addressed by either of them. Positive answers to this question were given by Dajczer-Florit-Tojeiro in 1998 for the ones called of hyperbolic type and by Dajczer-Florit in 2004 when of elliptic type which is the other possibility. In both cases the examples constructed are rather special. The main result of this paper yields an abundance of examples of hypersurfaces of either type and seems to point in the direction of a classification although that goal remains elusive.