Helicoidal Surfaces and Their Relationship to Bonnet Surfaces

Paul Bracken

doi:10.4236/apm.2017.71003

Abstract

An important question that arises is which surfaces in three-space admit a mean curvature preserving isometry which is not an isometry of the whole space. This leads to a class of surface known as a Bonnet surface in which the number of noncongruent immersions is two or infinity. The intention here is to present a proof of a theorem using an approach which is based on differential forms and moving frames and states that helicoidal surfaces necessarily fall into the class of Bonnet surfaces. Some other results are developed in the same manner.

Highlights

Surfaces that admit isometries which preserve principal curvatures have been studied since the time of O
This leads to a class of surface known as a Bonnet surface in which the number of noncongruent immersions is two or infinity
The intention here is to present a proof of a theorem using an approach which is based on differential forms and moving frames and states that helicoidal surfaces necessarily fall into the class of Bonnet surfaces

Summary

Introduction

Surfaces that admit isometries which preserve principal curvatures have been studied since the time of O. The surfaces in Euclidean space that admit a mean curvature preserving isometry which is not an isometry of the whole space form a special class of surface which has been studied by many people such as noted already Bonnet as well as Cartan and Chern [2] [3] [4] These surfaces may be broken up into three classes or types which can be described as follows: 1) There are surfaces of constant mean curvature other than the plane or sphere 2) There are certain surfaces of nonconstant mean curvature which admit a one-parameter family of geometrically distinct nontrivial isometries, and 3) There are surfaces of nonconstant mean curvature that admit a single nontrivial isometry which is unique up to an isometry of the entire space. Whereas a Bonnet surface of the third kind depends on four functions of one variable and has a greater degree of generality [10]

Structure Equations

Bonnet Surfaces

Helicoidal Surfaces

Main Theorems and Proofs