Hilbert forms for a Finsler metrizable projective class of sprays

TL;DR

This paper explores conditions under which a projective class of sprays can be derived from a Finsler metric, using Hilbert-type forms to establish equivalent criteria and addressing related geometric structures.

Contribution

It introduces Hilbert-type forms to formulate and prove equivalent necessary and sufficient conditions for Finsler metrizability of sprays, advancing the theoretical understanding of projective Finsler geometry.

Findings

Multiple equivalent conditions for Finsler metrizability are established.

Connections between path geometries and projective classes are clarified.

The role of Jacobi fields and totally-geodesic submanifolds is analyzed.

Abstract

The projective Finsler metrizability problem deals with the question whether a projective-equivalence class of sprays is the geodesic class of a (locally or globally defined) Finsler function. In this paper we use Hilbert-type forms to state a number of different ways of specifying necessary and sufficient conditions for this to be the case, and we show that they are equivalent. We also address several related issues of interest including path spaces, Jacobi fields, totally-geodesic submanifolds of a spray space, and the equivalence of path geometries and projective-equivalence classes of sprays.