An account on links between Finsler and Lorentz Geometries for Riemannian Geometers
arXiv:2203.13391 · doi:10.1007/978-3-031-39916-9_10
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.
arXiv comment39 pages, 8 figures
References in corpus (24)
- Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes
- Further results on the smoothability of Cauchy hypersurfaces and Cauchy time functions
- Stationary Metrics and Optical Zermelo-Randers-Finsler Geometry
- Globally hyperbolic spacetimes can be defined as "causal" instead of "strongly causal"
- Zermelo's problem: Optimal point-to-point navigation in 2D turbulent flows using Reinforcement Learning
- Causality theory for closed cone structures with applications
- Fermat Principle in Finsler Spacetimes
- On the energy functional on Finsler manifolds and applications to stationary spacetimes
- On the definition and examples of cones and Finsler spacetimes
- On the interplay between Lorentzian Causality and Finsler metrics of Randers type
- Standard static Finsler spacetimes
- Morse theory of causal geodesics in a stationary spacetime via Morse theory of geodesics of a Finsler metric
- On Finsler spacetimes with a timelike Killing vector field
- A general model for wildfire propagation with wind and slope
- Convex domains of Finsler and Riemannian manifolds
- Structure of globally hyperbolic spacetimes with timelike boundary
- Applications of cone structures to the anisotropic rheonomic Huygens' principle
- Chains in CR geometry as geodesics of a Kropina metric
- Wind Riemannian spaceforms and Randers metrics of constant flag curvature
- Static- and Stationary-complete Spacetimes: Algebraic and Causal Structures
- Can we make a Finsler metric complete by a trivial projective change?
- Connecting and closed geodesics of a Kropina metric
- Huygens' envelope principle in Finsler spaces and analogue gravity
- Lightlike hypersurfaces and time-minimizing geodesics in cone structures