A Note on Inextensible Flows of Curves on Oriented Surface
  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. 
Onder Gokmen Yildiz, Soley Ersoy, Melek Masal
math.DG