Genericity of nondegenerate geodesics with general boundary conditions

TL;DR

This paper proves that, generically, semi-Riemannian metrics on a manifold do not have degenerate geodesics satisfying certain boundary conditions, extending previous fixed-endpoint results to more general boundary scenarios.

Contribution

It generalizes the generic nondegeneracy of geodesics to boundary conditions defined by submanifolds, broadening the scope of previous fixed-endpoint results.

Findings

Generic metrics lack degenerate geodesics under specified boundary conditions.

Extension of fixed-endpoint results to submanifold boundary conditions.

Conditions under which metrics are C^k-generic without degenerate geodesics.

Abstract

Let M be a possibly noncompact manifold. We prove, generically in the C^k-topology (k=2,...,\infty), that semi-Riemannian metrics of a given index on M do not possess any degenerate geodesics satisfying suitable boundary conditions. This extends a result of Biliotti, Javaloyes and Piccione for geodesics with fixed endpoints to the case where endpoints lie on a compact submanifold P of the product MxM that satisfies an admissibility condition. Such condition holds, for example, when P is transversal to the diagonal of MxM. Further aspects of these boundary conditions are discussed and general conditions under which metrics without degenerate geodesics are C^k-generic are given.