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

TL;DR

This paper proves multiple existence results for orthogonal geodesic chords and brake orbits in certain Riemannian manifolds using Morse theory, expanding understanding of geodesic and orbit multiplicities.

Contribution

It introduces a new multiplicity theorem for orthogonal geodesic chords and brake orbits based on critical point estimates for Morse-even functions on spheres.

Findings

Multiple orthogonal geodesic chords exist in manifolds diffeomorphic to Euclidean balls.

Existence of multiple brake orbits in potential wells.

Application of Morse theory to geometric and dynamical problems.

Abstract

Using an estimate on the number of critical points for a Morse-even function on the sphere $\mathbb S^m$, $m\ge1$, we prove a multiplicity result for orthogonal geodesic chords in Riemannian manifolds with boundary that are diffeomorphic to Euclidean balls. This yields also a multiplicity result for brake orbits in a potential well.