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The Binet–Legendre Metric in Finsler Geometry

Vladimir S Matveev and Marc Troyanov

Geometry & Topology 16 (2012) 2135–2170

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.

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
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
Vladimir S Matveev
Mathematisches Institut, Friedrich-Schiller Universität Jena
Ernst-Abbe-Platz 2
D-07737 Jena
Marc Troyanov
Section de Mathématiques
École Polytechnique Féderale de Lausanne
Station 8
CH-1015 Lausanne