On the Berger conjecture for manifolds all of whose geodesics are closed

We define a Besse manifold as a Riemannian manifold (M, g) all of whose geodesics are closed. A conjecture of Berger states that all prime geodesics have the same length for any simply connected Besse manifold. We firstly show that the energy function on the free loop space of a simply connected Besse manifold is a perfect Morse–Bott function with respect to a suitable cohomology. Secondly we explain when the negative bundles along the critical manifolds are orientable. These two general results, then lead to a solution of Berger’s conjecture when the underlying manifold is a sphere of dimension at least four.