  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.
