Finsler metrizable isotropic sprays and Hilbert's fourth problem (Q2877687)

The present paper deals with the Finsler metrizability and Hilbert's fourth problem. It is well known that if a system of homogeneous second-order ordinary differential equations (spray) is metrizable by a Finsler function of scalar flag curvature then it must be necessarily isotropic. The main result of the present paper asserts that the isotropy condition, together with three other conditions concerning to Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. Also, the proof of the main result provides an algorithm to construct Finsler functions of scalar flag curvature, in the case when a given spray is metrizable and it is shown how the algorithm is used in order to construct solution to Hilbert's fourth problem. Very interesting examples that ilustrate their results are also presented.