Extendability of Kirchhoff elastic rods in complete Riemannian manifolds

The Kirchhoff elastic rod is a classical mathematical model of equilibrium configurations of thin elastic rods, and is defined to be a solution of the Euler-Lagrange equations associated to the energy with the effect of bending and twisting. We consider the initial-value problem for the Euler-Lagrange equations in a Riemannian manifold. In a previous paper, the author proved the existence and uniqueness of global solutions of the initial-value problem in the case where the ambient space is a space form. In the present paper, we extend this result to the case where the ambient space is a general complete Riemannian manifold. This implies that an arbitrary Kirchhoff elastic rod of finite length in a complete Riemannian manifold extends to that of infinite length.