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

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 navigati...

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Bibliographic Details
Published inarXiv.org
Main Authors Javaloyes, Miguel Ángel, Pendás-Recondo, Enrique, Sánchez, Miguel
Format Paper Journal Article
LanguageEnglish
Published Ithaca Cornell University Library, arXiv.org 24.10.2023
Subjects
Mathematics - Differential Geometry
Physics - General Relativity and Quantum Cosmology
Wave propagation
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Summary: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.
ISSN:2331-8422
DOI:10.48550/arxiv.2203.13391