Ruled and quadric surfaces of finite Chen-type

TL;DR

This paper classifies ruled and quadric surfaces in 3D Euclidean space that are of finite Chen-type with respect to the third fundamental form, identifying helicoids and spheres as unique solutions.

Contribution

It proves that only helicoids and spheres are ruled and quadric surfaces of finite III-type, respectively, in the context of Chen's finite type theory.

Findings

Helicoids are the only ruled surfaces of finite III-type.

Spheres are the only quadric surfaces of finite III-type.

The classification is based on the third fundamental form.

Abstract

In this paper, we study ruled surfaces and quadrics in the 3-dimensional Euclidean space which are of finite $III$-type, that is, they are of finite type, in the sense of B.-Y. Chen, with respect to the third fundamental form. We show that helicoids and spheres are the only ruled and quadric surfaces of finite $III$-type, respectively.