Morse theory on Hilbert manifolds

Summary

Morse theory on Hilbert manifolds extends classical Morse theory from compact manifolds to functions on infinite-dimensional Hilbert manifolds. Richard Palais and Stephen Smale replaced compactness of the manifold by a compactness condition on the function, the Palais–Smale condition. With it, sublevel sets change only at critical values, and passing a nondegenerate critical point of finite index $\lambda$ attaches a $\lambda$-cell. The main application is the study of geodesics as critical points of the energy on spaces of paths and loops.