Generic properties of semi-Riemannian geodesic flows

TL;DR

This paper establishes generic nondegeneracy properties of semi-Riemannian geodesic flows on manifolds, extending classical results to broader endpoint conditions and demonstrating typical non-conjugacy and non-focality.

Contribution

It proves a new version of the Bumpy Metric Theorem for semi-Riemannian manifolds and shows generic metrics lack degenerate geodesics under various endpoint conditions.

Findings

Generic metrics have no degenerate geodesics with specified endpoints.

Non-conjugacy between two points is a generic property.

Non-focality between points and submanifolds is typical.

Abstract

Let M be a possibly non compact smooth manifold. We study genericity in the C^k-topology (3<=k<=+infty) of nondegeneracy properties of semi-Riemannian geodesic flows on M. Namely, we prove a new version of the Bumpy Metric Theorem for a such M and also genericity of metrics that do not possess any degenerate geodesics satisfying suitable endpoints conditions. This extends results in the literature for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of MxM that satisfies an admissibility condition. Immediate consequences are generic non conjugacy between two points and non focality between a point and a submanifold (or also between two submanifolds).