Geodesic loops and orthogonal geodesic chords without self-intersections

TL;DR

This paper proves that for generic Riemannian metrics on certain manifolds, geodesic loops and orthogonal geodesic chords are typically without self-intersections, using perturbation and genericity techniques.

Contribution

It establishes generic conditions under which geodesic loops and orthogonal geodesic chords are free of self-intersections on manifolds with boundary.

Findings

Geodesic loops have no self-intersections for generic metrics.

Existence of multiple orthogonal geodesic chords without self-intersections.

Results apply to manifolds with convex boundary and dimension n ≥ 3.

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 and a genericity statement due to Bettiol and Giamb\`o and existence results for orthogonal geodesic chords by Giamb\`o, Giannoni, and Piccione.