Functions on the sphere with critical points in pairs and orthogonal geodesic chords (Q267503)

The main result of this paper establishes a multiplicity property for orthogonal geodesic chords in the framework of a Riemannian manifold with boundary that is diffeomorphic to Euclidean balls. The proof strongly relies on the estimate of the number of critical points for a Morse-even function on the Euclidean sphere. A direct consequence of the main result of the present paper yields a multiplicity property for brake orbits in a potential well. An index theorem for orthogonal geodesic chords plays a crucial role in the proof of this result. In the final part of this paper, using a stability result for focal points, the authors prove that the crossing time function is even-Morse, which provides a link between even-Morse functions and orthogonal geodesic chords.