A Finsler space is said to have reversible geodesics if for any of its oriented geodesic paths, the same path traversed in the opposite sense is also a geodesic. In [6] the conditions for a Randers space to have reversible geodesics have been found. The main goal of this paper is to find conditions for a Finsler space endowed with an (α, β)-metric to be with reversible geodesics or strictly reversible geodesics, respectively. Moreover, we obtain some new classes of (α, β)-metrics with reversible geodesics and show how new Finsler metrics with reversible geodesics can be constructed by means of a Randers change.

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.

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In [HS1], [MHSS] the L-duals of a Randers and Kropina space were studied. In this paper we shall discuss the L-dual of a Matsumoto space. The metric of this L-dual space is completely new and it brings a new idea about L-duality because the L-dual of Matsumoto metric can be given by means of four quadratic forms and 1-forms on T * M constructed only with the Riemannian metric coefficients, aij(x) and the 1-form coefficients bi(x).

We construct a concrete example of constant Gauss curvature K = 1 on the 2-sphere having all geodesics closed and of same length.