What is it about?

It is well known that a system of homogeneous second-order ordinary differential equations (spray) is necessarily isotropic in order to be metrizable by a Finsler function of scalar flag curvature. In our main theorem we show that the isotropy condition, together with three other conditions on the Jacobi endomorphism, characterize sprays that are metrizable by Finsler functions of scalar flag curvature. We call these theses conditions the scalar flag curvature (SFC) test. The proof of the main theorem provides an algorithm to construct the Finsler function of scalar flag curvature, in the case when a given spray is metrizable. Hilbert's fourth problem asks to determine the Finsler functions with rectilinear geodesics. A Finsler function that is a solution to Hilbert's fourth problem is necessarily of constant or scalar flag curvature. Therefore, we can use the constant flag curvature (CFC) or the SFC tests to decide when the projective deformations of a flat spray, which are isotropic, are metrizable by Finsler functions of constant or scalar flag curvature. We show how to use the algorithms provided CFC and SFC tests of to construct solutions to Hilbert's fourth problem.

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Why is it important?

The CFC and SFC tests provide an answer to the question if a given isotropic spray is Finsler metrizable. In the affirmative case it also provides the Finsler function that metricises it.

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This page is a summary of: FINSLER METRIZABLE ISOTROPIC SPRAYS AND HILBERT’S FOURTH PROBLEM, Journal of the Australian Mathematical Society, May 2014, Cambridge University Press,
DOI: 10.1017/s1446788714000111.
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