Geodesic loops and orthogonal geodesic chords without self-intersections

Abstract

We show that for a generic Riemannian metric on a compact manifold of dimension $n\ge 3$ all geodesic loops based at a fixed point have no self-intersections. We also show that for an open and dense subset of the space of Riemannian metrics on an $n$-disc with $n\ge 3$ and with a strictly convex boundary there are $n$ geometrically distinct orthogonal geodesic chords without self-intersections. We use a perturbation result for intersecting geodesic segments of the author Rademacher (2024) and a genericity statement due to Bettiol and Giambò (2010) and existence results for orthogonal geodesic chords by Giambò et al. (2018).