Traveling along horizontal broken geodesics of a homogeneous Finsler submersion

In this paper, we discuss how to travel along horizontal broken geodesics of a homogeneous Finsler submersion, i.e., we study, what in Riemannian geometry was called by Wilking, the dual leaves. More precisely, we investigate the attainable sets $A_q\left(C\right)$ of the set of analytic vector fields $C$ determined by the family of horizontal unit geodesic vector fields $C$ to the fibers $F=\left\{{\rho }^{-1}\right(c\left)\right\}$ of a homogeneous 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 $C$ 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 $O\left(q\right)$ of $C$ 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.