The Binet-Legendre Metric in Finsler Geometry
Vladimir S. Matveev, Marc Troyanov

TL;DR
This paper introduces the Binet-Legendre metric, a stable and smooth Riemannian metric associated with Finsler metrics, enabling new solutions and classifications in Finsler geometry, even under relaxed smoothness conditions.
Contribution
It defines the Binet-Legendre metric, demonstrating its stability and utility in solving various Finsler geometric problems and generalizing classical results.
Findings
Solved a question of M. Matsumoto on Minkowski spaces
Classified all conformally flat Finsler manifolds
Extended Wang's theorem on isometry groups
Abstract
For every Finsler metric we associate a Riemannian metric (called the Binet-Legendre metric). The transformation is -stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric also behaves nicely under conformal or bilipshitz deformation of the Finsler metric . These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Finslerian geometric problems. We also generalize and give new and shorter proofs of a number of known results. In particular we answer a question of M. Matsumoto about local conformal mapping between two Minkowski spaces, we describe all possible conformal self maps and all self similarities on a Finsler manifold. We also classify all compact conformally flat Finsler manifolds, solve a conjecture of S. Deng and Z. Hou on the…
Peer Reviews
No public reviews on file for this paper yet. If you reviewed it on a platform where reviews are public (OpenReview, ICLR, NeurIPS, ICML), you can paste yours below so the community can read it here.
Videos
No videos yet. Explain this paper in a talk, walkthrough, or lecture? Add one.
