Title: Genericity of nondegenerate critical points and Morse geodesic functionals

Authors: Leonardo Biliotti, Miquel Angel Javaloyes and Paolo Piccione

Issue: Volume 58 (2009), Issue 4, 1797-1830

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 [B. White, \textit{The space of minimal submanifolds for varying Riemannian metrics}, Indiana Univ. Math. J. \textbf{40} (1991), 161-200]. 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.