Tubes of finite $II$-type Gauss map

TL;DR

This paper classifies finite type Gauss map surfaces in Euclidean space, focusing on tubes, and demonstrates that tubes have an infinite type Gauss map related to the second fundamental form.

Contribution

It introduces a family of tube surfaces in Euclidean space and proves their Gauss map is of infinite type, extending the classification of finite type Gauss maps.

Findings

Tubes in Euclidean space have Gauss maps of infinite type.

The classification of finite type Gauss maps is extended to include tubes.

Tubes' Gauss maps correspond to the second fundamental form, indicating their infinite type.

Abstract

In this paper, we continue the classification of finite type Gauss map surfaces in the 3-dimensional Euclidean space $\mathbb{E}^{3}$. We present an important family of surfaces, namely, tubes in $\mathbb{E}^{3}$. We show that the Gauss map of a tube is of an infinite type corresponding to the second fundamental form.