Some properties of surfaces of finite III-type

TL;DR

This paper explores properties of surfaces in Euclidean space related to the third fundamental form, introduces finite Chen type surfaces of revolution with nonzero Gauss curvature, and examines specific cases with constant principal curvature sums.

Contribution

It investigates relations involving Laplace operators of the third fundamental form and introduces a new class of finite Chen type surfaces of revolution.

Findings

Relations between Laplace operators and the third fundamental form

Definition of finite Chen type surfaces of revolution with nonzero Gauss curvature

Special case of surfaces with constant sum of principal curvature radii

Abstract

In this paper, we firstly investigate some relations regarding the first and the second Laplace operators corresponding to the third fundamental form III of a surface in the Euclidean space E3. Besides, we introduce the finite Chen type surfaces of revolution with nonvanishing Gauss curvature with respect to the third fundamental form. We present a special case of this family of surfaces of revolution in E3, namely, surfaces of revolution with R is constant, where R denotes the sum of the radii of the principal curvature of a surface.