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 (C) of the set of analytic vector fields C determined by the family of horizontal unit geodesic vector fields C to the fibers F = { ρ − 1 (c) } of a homogeneous analytic Finsler submersion ρ : M → 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 (q) 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. [ABSTRACT FROM AUTHOR]