An account on links between Finsler and Lorentz Geometries for Riemannian Geometers

Abstract

Some links between Lorentz and Finsler geometries have been developed in the last years, with applications even to the Riemannian case. Our purpose is to give a brief description of them, which may serve as an introduction to recent references. As a motivating example, we start with Zermelo navigation problem, where its known Finslerian description permits a Lorentzian picture which allows for a full geometric understanding of the original problem. Then, we develop some issues including: (a) the accurate description of the Lorentzian causality using Finsler elements, (b) the non-singular description of some Finsler elements (such as Kropina metrics or complete extensions of Randers ones with constant flag curvature), (c) the natural relation between the Lorentzian causal boundary and the Gromov and Busemann ones in the Finsler setting, and (d) practical applications to the propagation of waves and firefronts.

The Finsler Spacetime Condition for (α,β)-Metrics and Their Isometries

“…All the above-mentioned applications require Finsler metrics of Lorentzian signature. While there exists a rich literature on positive definite Finsler metrics (and in particular, on (α, β)-ones [25][26][27][28][29][30]), Lorentzian Finsler geometry is by far least understood and investigated [31]. In this paper, we study for the first time in full generality two questions about Lorentzian (α, β)-Finsler structures, which have been only partially tackled in the literature (mostly only for very particular cases):…”