We study geodesics of the H1 Riemannian metric (see article for equation) on the space of inextensible curves (see article for equation). This metric is a regularization of the usual L2 metric on curves, for which the submanifold geometry and geodesic equations have been analyzed already. The H1 geodesic equation represents a limiting case of the Pochhammer-Chree equation from elasticity theory. We show the geodesic equation is C∞ in the Banach topology C1 ([0,1], R2), and thus there is a smooth Riemannian exponential map. Furthermore, if we hold one of the curves fixed, we have global-in-time solutions. We conclude with some surprising features in the periodic case, along with an analogy to the equations of incompressible fluid mechanics.
Preston, Stephen C. and Saxton, Ralph, "An H1 Model for Inextensible Strings" (2012). Mathematics Faculty Publications. Paper 19.