Two dimensional $(\alpha,\beta)$-metrics with reversible geodesics

Abstract

We study the necessary and sufficient conditions for a Finsler surface with (α, β)-metrics to be with reversible geodesics. We show that such a Finsler structure is with reversible geodesics if and only if it is a Randers change of an absolute homogeneous Finsler metric by a closed one-form.

Metric structures associated to Finsler metrics

“…We have (see [MSS10], [MSS13], [SS12] and references herein) The intuitive meaning of the Randers change…”

Two dimensional $(\alpha,\beta)$-metrics with reversible geodesics

“…Straightforward computations give immediately the relations between the directional derivatives of p and r with respect to the Riemannian coframing {α 1 , α 2 , α 3 } and the partial derivatives with respect to the natural coordinates (x 1 , x 2 , t). Since these computations are quite long and annoying we decided to put them in a preliminary version of this paper [MSS2] available on arxiv.org.…”

“…Straightforward computations (that can be found in [MSS2]) lead us to Theorem 4.2. The necessary and sufficient condition for the Finsler structures F (x, y) and F (x, y) = F (x, −y) to be projectively equivalent is…”

On the reversible geodesics of a Finsler space with special (α,β )-metric

“…In [1], Crampin discusses Randers space with reversible geodesics. In [4,5], Masca et. al discuss reversible geodesics with (α, β)-metrics and two dimensional (α, β)-metrics with reversible geodesic respectively.…”