Timelike Tangent Developable Surfaces and Bonnet Surfaces

Soley Ersoy,Kemal Eren

doi:10.1155/2016/6837543

Abstract

A criterion was given for a timelike surface to be a Bonnet surface in 3-dimensional Minkowski space by Chen and Li, 1999. In this study, we obtain a necessary and sufficient condition for a timelike tangent developable surface to be a timelike Bonnet surface by the aid of this criterion. This is examined under the condition of the curvature and torsion of the base curve of the timelike developable surface being nonconstant. Moreover, we investigate the nontrivial isometry preserving the mean curvature for a timelike flat helicoidal surface by considering the curvature and torsion of the base curve of the timelike developable surface as being constant.

Highlights

Surfaces which admit a one-parameter family of isometric deformations preserving the mean curvature are called Bonnet surfaces
Characterization for isometric deformation preserving the principal curvatures of surfaces was obtained by the aid of differential forms by Chern in [4]
The geometric characterizations of helicoidal surfaces of constant mean curvature, helicoidal surfaces as Bonnet surfaces, and tangent developable surfaces as Bonnet surfaces were studied by Roussos in [5], [6] and [7], respectively

Summary

Introduction

Surfaces which admit a one-parameter family of isometric deformations preserving the mean curvature are called Bonnet surfaces. It was proved that any Bonnet surface of nonconstant mean curvature depends on six arbitrary constants [3]. Characterization for isometric deformation preserving the principal curvatures of surfaces was obtained by the aid of differential forms by Chern in [4]. Roussos obtained a characterization for isometric deformation preserving the mean curvature by using the method of Chern. Soyucok proved that 3-dimensional hyperspace in 4-dimensional space is Bonnet surface if and only if hypersurface has orthogonal net [9]. Bagdatlı and Soyucok studied hypersurfaces preserving the mean curvature and proved that a hypersurface in Rn+1 is Bonnet surface if and only if hypersurface has orthogonal A-net [10]. Chen and Li studied 3-dimensional Minkowski space and classified timelike Bonnet surfaces [11]

Preliminaries

Timelike Bonnet Surfaces in Minkowski Space

Timelike Tangent Developable Surfaces