Intrinsic equations for a relaxed elastic line on an oriented surface

Hilbert and Cohn-Vossen 2, p. 221] incorrectly suggested a flexible knitting needle, constrained to conform to a surface, as one model for a geodesic on a surface. This model actually gives a relaxed elastic line on the surface, and is not generally a geodesic unless the surface lies in a plane or on a sphere.In this paper we derive the intrinsic equations for a relaxed elastic line on an oriented surface. This formulation should give a more direct and more geometric approach to questions concerning relaxed elastic lines on a surface. We apply this formulation to give alternate proofs of some results of 3] found by the less direct method of Lagrange multipliers and to give additional results about relaxed elastic lines on various surfaces. For further considerations of a relaxed elastic line on a surface as a model of the DNA molecule, see 3].Partially supported by NIH grant GM 36284-01.