On the projective theory of sprays with applications to Finsler geometry

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 - Szabó theorem on isotropic Finsler manifolds, the Finslerian Schur lemma, and the uniqueness of the Berwald connection of a Finsler manifold.