A Global Approach to the Theory of Connections in Finsler Geometry

TL;DR

This paper develops a global, coordinate-free framework in Finsler geometry to prove the existence and uniqueness of key connections, linking local classical results with a comprehensive intrinsic approach.

Contribution

It introduces an intrinsic, global method for analyzing Finsler connections, unifying and extending classical local results with coordinate-free proofs.

Findings

Intrinsic proofs of existence and uniqueness of Chern and Hashiguchi connections

Explicit global expressions relating these connections to the Cartan connection

Coincidence of associated semispray and nonlinear connection with canonical structures

Abstract

Adopting the pullback approach to global Finsler geometry, the aim of the present paper is to provide intrinsic (coordinate-free) proofs of the existence and uniqueness theorems for the Chern (Rund) and Hashiguchi connections on a Finsler manifold. To accomplish this, we introduce and investigate the notions of semispray and nonlinear connection associated with a given regular connection, in the pullback bundle. Moreover, it is shown that for the the Chern (Rund) and Hashiguchi connections, the associated semispray coincides with the canonical spray and the associated nonlinear connection coincides with the Barthel connection. Explicit intrinsic expressions relating these connections and the Cartan connection are deduced. Although our investigation is entirely global, the local expressions of the obtained results, when calculated, coincide with the existing classical local results. We provide, for the sake of completeness and for comparison reasons, two appendices, one of them presenting a global survey of canonical linear connections in Finsler geometry and the other presenting a local survey of our global approach.