  A conjecture of Berger states that, for any simply connected Riemannian
manifold all of whose geodesics are closed, all prime geodesics have the same
length. We firstly show that the energy function on the free loop space of such
a 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.
