Morse theory for Riemannian geodesics without nondegeneracy assumptions (Q1130198)

Let \(M\) be a Hilbert manifold and \(f\in C^2(M,\mathbb R)\) be bounded from below, satisfying the Palais-Smale condition and such that the Hessian of \(f\) at a critical point is Fredholm of index \(0\). The critical points need not be isolated. In this situation the authors define a generalized Morse index of \(f\). The definition is based on considering the set of classical Morse polynomials of certain perturbations of \(f\) which are Morse functions. The classical Morse inequalities generalize immediately. The abstract theory is applied to the theory of geodesics joining two points on a finite-dimensional Riemannian manifold. Here \(f\) is the Dirichlet integral. The authors also introduce a geometric Morse index of \(f\) based on the notion of conjugate points for critical points of perturbations of \(f\).