# Tubes of finite Chen-type

## TL;DR

This paper investigates surfaces in three-dimensional space that have finite type with respect to the third fundamental form, specifically analyzing tubes and demonstrating they are of infinite III-type.

## Contribution

The paper introduces the family of tubes in E3 and proves they are of infinite III-type, expanding understanding of surface classifications.

## Key findings

- Tubes in E3 are of infinite III-type.
- Surfaces of finite III-type are characterized in the study.
- The paper provides new insights into the classification of surfaces based on fundamental forms.

## Abstract

In this paper, we consider surfaces in the 3-dimensional Euclidean space E3 which are of finite III-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the third fundamental form. We present an important family of surfaces, namely, tubes in E3 .We show that tubes are of infinite III-type.