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.

39 pages, 8 figures

References in corpus (8)

• Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry
• Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
• A general model for wildfire propagation with wind and slope
• Applications of cone structures to the anisotropic rheonomic Huygens' principle
• Static- and Stationary-complete Spacetimes: Algebraic and Causal Structures
• Connecting and closed geodesics of a Kropina metric
• Huygens' envelope principle in Finsler spaces and analogue gravity
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Cited by in corpus (2)

• Globally hyperbolic spacetimes: slicings, boundaries and counterexamples
• On the metrizability of $m$-Kropina spaces with closed null 1-form