On the projective theory of sprays with applications to Finsler geometry

TL;DR

This paper provides a comprehensive, coordinate-free analysis of sprays and Finsler geometry, deriving new results on tensor properties, metrizability criteria, and classical theorems with detailed proofs and applications.

Contribution

It introduces new insights into the direction-independence of tensors, characterizes Finsler-metrizability, and offers novel proofs of classical results within a coordinate-free framework.

Findings

Vanishing projected Berwald tensor implies a Berwald manifold in dimension ≥3

Criteria for Finsler-metrizability and projective Finsler-metrizability established

New proofs of classical theorems like the Berwald - del Castillo - Szabó theorem

Abstract

Based on a self-contained, coordinate-free exposition of the necessary concepts and tools of spray and Finsler geometry (with detailed proofs), we derive new results among others on the consequences of the direction-independence of the Landsberg tensor and the stretch tensor of a Finsler manifold. We show that an at least 3-dimensional Finsler manifold with vanishing projected Berwald tensor is a Berwald manifold. To obtain consequences in two dimensions, we transcript Berwald's classical theory on 2-dimensional Finsler manifolds in our setup. We prove criteria and necessary conditions for Finsler-metrizability and projective Finsler-metrizability of a spray. We present new proofs for such classical results as the Berwald - del Castillo - Szab\'o theorem on isotropic Finsler manifolds, the Finslerian Schur lemma, and the uniqueness of the Berwald connection of a Finsler manifold.