  This paper studies a variational obstacle avoidance problem on complete
Riemannian manifolds. That is, we minimize an action functional, among a set of
admissible curves, which depends on an artificial potential function used to
avoid obstacles. In particular, we generalize the theory of bi-Jacobi fields
and biconjugate points and present necessary and sufficient conditions for
optimality. Local minimizers of the action functional are divided into two
categories and subsequently classified, with local uniqueness results obtained
in both cases.
