The exponential map on the free loop space is Fredholm
In the main result of the paper, sufficient conditions for the exponential map of an infinite-dimensional Riemannian (Hilbert) manifold to be a nonlinear Fredholm map of index zero are given. This result can be applied to the free loop space of a finite-dimensional manifold. In particular, it follows that for a conjugate point the derivative of the exponential map is neither injective nor surjective which answers a question posed by W. Klingenberg. Another consequence is that conjugate points on the free loop space cannot cluster along finite geodesic arcs.
scientific article; zbMATH DE number 1123716