# Ruled surfaces of finite type with respect to the second fundamental form

## TL;DR

This paper investigates ruled surfaces in 3D Euclidean space to determine if they are of finite type with respect to the second fundamental form, concluding that all ruled surfaces are of infinite II-type.

## Contribution

It establishes that ruled surfaces in Euclidean space are of infinite II-type, expanding understanding of surface classifications in differential geometry.

## Key findings

- Ruled surfaces are of infinite II-type.
- The study focuses on surfaces without parabolic points.
- Results contribute to the classification of surfaces by fundamental forms.

## Abstract

In this article, we consider surfaces in the 3-dimensional Euclidean space E3 without parabolic points which are of finite II-type, that is, they are of finite type, in the sense of B.-Y. Chen, corresponding to the second fundamental form. We study an important family of surfaces, namely, ruled surfaces in E3. We show that ruled surfaces are of infinite II-type.