A Note on Inextensible Flows of Curves on Oriented Surface

TL;DR

This paper develops a mathematical framework for inextensible curve flows on oriented surfaces in three-dimensional space, deriving conditions using differential equations involving geodesic curvature and torsion.

Contribution

It introduces a general formulation and necessary conditions for inextensible flows of curves on oriented surfaces, including special cases.

Findings

Derived PDE conditions for inextensible curve flows.

Identified special cases of inextensible curves.

Provided mathematical characterization of curve behavior on surfaces.

Abstract

In this paper, the general formulation for inextensible flows of curves on oriented surface in $\mathbb{R}^3 $ is investigated. The necessary and sufficient conditions for inextensible curve flow lying an oriented surface are expressed as a partial differential equation involving the geodesic curvature and the geodesic torsion. Moreover, some special cases of inextensible curves on oriented surface are given.