Genericity of nondegenerate critical points and Morse geodesic functionals

TL;DR

This paper proves that, under broad conditions, nondegenerate critical points are generic in certain variational problems on infinite-dimensional manifolds, with applications to geodesic properties in various semi-Riemannian and Lorentzian geometries.

Contribution

It establishes an abstract genericity theorem for nondegenerate critical points using the Sard--Smale theorem and applies it to geodesic problems in semi-Riemannian and Lorentzian manifolds.

Findings

Generic metrics have no degenerate geodesics between fixed points.

The nondegeneracy condition is generic in stationary Lorentzian manifolds.

Applications include semi-Riemannian and globally hyperbolic Lorentzian geometries.

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. Based on an idea of B. White, we prove an abstract genericity result that employs the infinite dimensional Sard--Smale theorem. 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.