Let M be an oriented Riemannian manifold all of whose geodesic are closed, then the index of any closed geodesic in the free loop space of M is odd if M is even dimensional, and it is even if M is odd dimensional. Using this result, B. Wilking proved that any Riemannian submersion from the round sphere with connected fibers of positive dimension is metric congruent to a generalized Hopf fibration.