A Condition for Classical Elastic Curves on Surface

Abstract

In this paper, we consider two fixed points p to q on a Riemannian surface M in 3-dimensional Euclidean space. We obtain a condition for classical elastic curves with in the family of all curves from p to q on M. We also prove that this condition can be expressed in terms of the curvature functions. The condition is realized for curves whose geodesic and normal curvature functions are both constant.