  This paper studies sufficient conditions in 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. We provide necessary and sufficient
conditions under which the resulting critical points, the so-called modified
Riemannian cubics, are local minimizers. We then study the theory of reduction
by symmetries of sufficient conditions for optimality in variational obstacle
avoidance problems on Lie groups endowed with a left-invariant metric. This
amounts to left-translating the Bi-Jacobi fields described to the Lie algebra,
and studying the corresponding bi-conjugate points. New conditions are provided
in terms of the invertibility of a certain matrix.
