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

TL;DR

This paper explores the connections between Finsler and Lorentz geometries, illustrating their applications to Riemannian geometry through examples like the Zermelo navigation problem and causality analysis.

Contribution

It provides a comprehensive overview of recent developments linking Lorentzian and Finsler geometries, including applications to wave propagation and geometric boundaries.

Findings

Finslerian description enables Lorentzian causality analysis.

Non-singular Finsler elements can be effectively described.

Connections between Lorentzian and Finsler boundaries are established.

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.