Second order optimality conditions for a higher order variational problem on a Riemannian manifold

In this paper, we derive second order optimality conditions for a higher order variational problem on a general Riemannian manifold, which can be viewed as an extension of the minimizing acceleration problem in Euclidean space and yields the geometric generalization of the classical cubic polynomials. This continues the work initiated by Crouch and Silva Leite (1995). In particular, we define conjugate points and prove a necessary and sufficient condition for optimality, in the absence of such singularities.