Traveling along horizontal broken geodesics of a homogenous Finsler submersion

TL;DR

This paper explores how to travel along horizontal broken geodesics in homogeneous Finsler submersions, establishing conditions under which attainable sets coincide with orbits and demonstrating travel between points on compact manifolds with positive flag curvature.

Contribution

It extends Wilking's results on dual leaves to Finsler geometry, analyzing attainable sets and orbits in the context of homogeneous Finsler submersions with positive flag curvature.

Findings

Attainable sets coincide with orbits on compact manifolds with embedded orbits.

Positive flag curvature implies the entire manifold is an attainable set from any point.

Wilking's results on dual leaves are generalized to Finsler geometry.

Abstract

In this paper, we discuss how to travel along horizontal broken geodesics of a homogenous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $\mathcal{A}_{q}(\mathcal{C})$ of the set of analytic vector fields $\mathcal{C}$ determined by the family of horizontal unit geodesic vector fields $\mathcal{C}$ to the fibers $\mathcal{F}=\{\rho^{-1}(c)\}$ of a homogenous analytic Finsler submersion $\rho: M\to B$. Since reverse of geodesics don't need to be geodesics in Finsler geometry, one can have examples on non compact Finsler manifolds $M$ where the attainable sets (the dual leaves) are no longer orbits or even submanifolds. Nevertheless we prove that, when $M$ is compact and the orbits of $\mathcal{C}$ are embedded, then the attainable sets coincide with the orbits. Furthermore, if the flag curvature is positive then $M$ coincides with the attainable set of each point. In other words, fixed two points of $M$, one can travel from one point to the other along horizontal broken geodesics. In addition, we show that each orbit $\mathcal{O}(q)$ of $\mathcal{C}$ associated to a singular Finsler foliation coincides with $M$, when the flag curvature is positive, i.e, we prove Wilking's result in Finsler context. In particular we review Wilking's transversal Jacobi fields in Finsler case.