Genericity of Nondegenerate Critical Points and Morse Geodesic Functionals

Abstract

We consider a family of variational problems on a Hilbert manifold parameterized by an open subset of a Banach manifold, and we discuss the genericity of the nondegeneracy condition for the critical points. Using classical techniques, we prove an abstract genericity result that employs the infinite dimensional Sard-Smale theorem, along the lines of an analogous result of B. White [29]. Applications are given by proving the genericity of metrics without degenerate geodesics between fixed endpoints in general (non compact) semi-Riemannian manifolds, in orthogonally split semi-Riemannian manifolds and in globally hyperbolic Lorentzian manifolds. We discuss the genericity property also in stationary Lorentzian manifolds.