Curvature equivalence for Legendre curves in the unit tangent bundle over Euclidean plane

TL;DR

This paper introduces the concept of curvature equivalence for Legendre curves in the unit tangent bundle over the Euclidean plane, providing classifications and exploring properties of these curves.

Contribution

It defines a new equivalence relation called curvature equivalence and offers local and global classifications of Legendre curves under this relation.

Findings

Established existence and uniqueness theorems for Legendre curve curvature.

Developed local and global classification results for Legendre curves.

Analyzed properties of curvature equivalence for Legendre curves.

Abstract

The Legendre curve in the unit tangent bundle over Euclidean plane is a plane curve with a moving frame. We have the (Legendre) curvature of the Legendre curve, and the existence and uniqueness theorems for the curvature are valid. In this paper, we introduce an equivalence relation for Legendre curves called the curvature equivalence. We investigate properties of the curvature equivalence and give local and global classifications of Legendre curves under the curvature equivalence.