Existence of orthogonal geodesic chords on Riemannian manifolds with concave boundary and homeomorphic to the N-dimensional disk

TL;DR

This paper proves the existence of orthogonal geodesic chords on certain Riemannian manifolds with concave boundaries, linking geometric analysis to Hamiltonian dynamics and the Seifert conjecture.

Contribution

It provides a new proof of orthogonal geodesic chords existence on manifolds homeomorphic to disks with concave boundaries, connecting geometry and dynamical systems.

Findings

Existence of orthogonal geodesic chords on manifolds with concave boundary

Connection between geometric boundary conditions and Hamiltonian systems

Implications for the Seifert conjecture and brake orbits

Abstract

In this paper we give a proof of the existence of an orthogonal geodesic chord on a Riemannian manifold homeomorphic to a closed disk and with concave boundary. This kind of study is motivated by the link of the multiplicity problem with the famous Seifert conjecture (formulated in 1948) about multiple brake orbits for a class of Hamiltonian systems at a fixed energy level.