#### Volume 16, issue 4 (2012)

 Recent Issues
 The Journal About the Journal Subscriptions Editorial Board Editorial Interests Editorial Procedure Submission Guidelines Submission Page Author Index To Appear ISSN (electronic): 1364-0380 ISSN (print): 1465-3060
The Binet–Legendre Metric in Finsler Geometry

### Vladimir S Matveev and Marc Troyanov

Geometry & Topology 16 (2012) 2135–2170
##### Abstract

For every Finsler metric $F$ we associate a Riemannian metric ${g}_{F}$ (called the Binet–Legendre metric). The Riemannian metric ${g}_{F}$ behaves nicely under conformal deformation of the Finsler metric $F$, which makes it a powerful tool in Finsler geometry. 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, we solve a conjecture of S Deng and Z Hou on the Berwaldian character of locally symmetric Finsler spaces, and extend a classic result by H C Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions.

Most proofs in this paper go along the following scheme: using the correspondence $F↦{g}_{F}$ we reduce the Finslerian problem to a similar problem for the Binet–Legendre metric, which is easier and is already solved in most cases we consider. The solution of the Riemannian problem provides us with the additional information that helps to solve the initial Finslerian problem.

Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced by a weaker partial smoothness, a notion we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.

##### Keywords
Finsler metrics, conformal transformations, conformal invariants, locally symmetric spaces, Berwald spaces, Killing vector fields
##### Mathematical Subject Classification 2010
Primary: 53C60, 58B20
Secondary: 53C35, 30C20, 53A30
##### Publication
Received: 19 January 2012
Revised: 15 May 2012
Accepted: 9 July 2012
Published: 13 November 2012
Proposed: Dmitri Burago
Seconded: Yasha Eliashberg, Jean-Pierre Otal
##### Authors
 Vladimir S Matveev Mathematisches Institut, Friedrich-Schiller Universität Jena Ernst-Abbe-Platz 2 D-07737 Jena Germany http://users.minet.uni-jena.de/~matveev/ Marc Troyanov Section de Mathématiques École Polytechnique Féderale de Lausanne Station 8 CH-1015 Lausanne Switzerland http://personnes.epfl.ch/marc.troyanov