Singularities of Developable Surfaces

Summary

Dedicated to the 60th birthday of Professor C.T.C. Wall.

Abstract

In this survey article we explain recent results concerning singularities appearing on the tangent developables of space curves via the notion of projective duality. First we examine, from our viewpoint, the classical local classification and the Hartman-Nirenberg theorem on developable surfaces. Then we introduce results on the local differentiable and topological classification of tangent developables of space curves. Lastly we present some related problems and questions, for instance concerning finite determinacy.

Introduction

A smooth surface in Euclidean three space is called a developable surface if its Gaussian curvature vanishes everywhere. By Gauss' fundamental theorem, a developable surface is locally isometric to a planar domain. This motivates the name “developable” (see [58]). Let z = f(x,y) be a smooth surface. If we set r = z_xx, s = z_xy, t = z_yy, then the surface is a developable surface if and only if the Monge-Ampère equation rt – s² = 0 is satisfied on the surface [47], [34].

Developable surfaces are well-known objects from the last century. For instance, analytic developable surfaces are locally classified into three types: cones, cylinders and tangent developables. In the smooth case, we can rather freely change the three types through planar trapezia to make various (not necessarily analytic) non-singular portions of smooth developable surfaces.