Linear Weingarten Helicoidal Surfaces in Isotropic Space

Dae Yoon,Jae Lee

doi:10.3390/sym8110126

Abstract

In the present paper, we study helicoidal surfaces in the three-dimensional isotropic space I 3 and construct helicoidal surfaces satisfying a linear equation in terms of the Gaussian curvature and the mean curvature of the surface.

Highlights

Introduced in 1861 [1], a Weingarten surface in the Euclidean three-dimensional space E3 is a surface M, whose mean curvature H and Gaussian curvature K satisfy a non-trivial relationΦ( H, K ) = 0
The class of Weingarten surfaces is remarkably large, and it consists of intriguing surfaces in the Euclidean space: the constant mean curvature surfaces, the constant Gaussian curvature surfaces and all rotational surfaces
To define helicoidal surfaces in an isotropic space I 3, we consider a curve γ lying in the isotropic xz-plane or yz-plane without loss of generality

Summary

Introduction

Introduced in 1861 [1], a Weingarten surface in the Euclidean three-dimensional space E3 is a surface M, whose mean curvature H and Gaussian curvature K satisfy a non-trivial relation We study helicoidal surfaces in an isotropic three-dimensional space. The main purpose of this paper is to construct helicoidal surfaces in an isotropic three-dimensional space satisfying the linear Weingarten relation (1).

Objectives

Results

Conclusion