Conditional extremal curves in a complete Riemannian manifold M are defined as the critical
points of the squared L2 distance between the tangent vector field of a curve and a so-called
prior vector field. We prove that this L2 distance satisfies the Palais–Smale condition on the
space of absolutely continuous curves joining two submanifolds of M, and thus establish the
existence of critical points. We also prove a Morse index theorem in the case where the two
submanifolds are single points, and use the Morse inequalities to place lower bounds on the …