Geometry & Topology

The Binet–Legendre Metric in Finsler Geometry

Vladimir S Matveev and Marc Troyanov

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For every Finsler metric F we associate a Riemannian metric gF (called the Binet–Legendre metric). The Riemannian metric gF 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 FgF 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.

Article information

Geom. Topol., Volume 16, Number 4 (2012), 2135-2170.

Received: 19 January 2012
Revised: 15 May 2012
Accepted: 9 July 2012
First available in Project Euclid: 20 December 2017

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Mathematical Reviews number (MathSciNet)

Zentralblatt MATH identifier

Primary: 53C60: Finsler spaces and generalizations (areal metrics) [See also 58B20] 58B20: Riemannian, Finsler and other geometric structures [See also 53C20, 53C60]
Secondary: 53C35: Symmetric spaces [See also 32M15, 57T15] 30C20: Conformal mappings of special domains 53A30: Conformal differential geometry

Finsler metrics conformal transformations conformal invariants locally symmetric spaces Berwald spaces Killing vector fields


Matveev, Vladimir S; Troyanov, Marc. The Binet–Legendre Metric in Finsler Geometry. Geom. Topol. 16 (2012), no. 4, 2135--2170. doi:10.2140/gt.2012.16.2135.

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