Morse theory for Riemannian geodesics without nondegeneracy assumptions

Abstract

Let f ∈ C(M,R) be a functional defined on a Hilbert manifold M. It is well known that if f is a Morse functional (i.e. every critical point of f is nondegenerate) and f satisfies the so called Palais–Smale condition, the Morse relations hold. More precisely, let x ∈ M be a critical point of f , and m(x, f) denote the Morse index at x (i.e. the maximal dimension of the subspaces of TxM where the hessian at x is negative definite). The polynomial defined by mλ(f) = ∑