On the projective theory of sprays with applications to Finsler geometry

Details On the projective theory of sprays with applications to Finsler geometry On the projective theory of sprays with applications to Finsler geometry

2009-08-30

arxiv.org/abs/0908.4384v2

Mathematics

Differential Geometry

Final version of the author's PhD Thesis

Scientific paper

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.

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