Morse theory for the free loop space of a projective space

The free loop space on a compact Riemannian manifold can be made into a Hilbert manifold. The energy integral is a smooth function on this Hilbert manifold, which calculates the energy of closed loops.
Its critical points are the closed geodesics on the Riemannian manifold, and the critical points of the same energy level form a critical submanifold of the free loop space. There is a so called negative vector
bundle over each critical submanifold and the associated disk bundles enter in a Morse decomposition of the free loop space.

For projective spaces over the complex numbers, the quaternions or the Cayley numbers, Wilhelm Klingenberg and Wolfgang Ziller used this Morse theory approach to compute the homology of the associated free loop spaces.  Marcel Bökstedt and I have done computations for equivariant cohomology
with respect to the rotation action of the circle group.

The aim of this project is to obtain a simple description of the  negative bundles for projective spaces, and to examine the implications of such a description in equivariant cohomology.