Hyperelastic curves along immersions

Abstract

Hyperelastic curves in a Riemannian manifold are solutions of a constrained variational problem and characterized by Euler-Lagrange equations. We study the effect of hyperelastic curves on the geometry of isometric immersions. We investigate the relation between hyperelastic curves and umbilical submanifolds and apply the results to analyze classical elastic curves. The case of a Riemannian manifold with constant sectional curvature is also discussed and some applications are presented for illustrating the results.