THE BINET-LEGENDRE ELLIPSOID IN FINSLER GEOMETRY

In this paper we introduce a new construction that associates a Riemannian metric gF (called the Binet-Legendre metric) to a given Finsler metric F on a smooth man- ifold M. The transformation F 7→ gF is C 0 -stable and has good smoothness properties, in contrast to previous constructions. The Riemannian metric gF also behaves nicely under conformal or bilipshitz deformation of the Finsler metric F. These properties makes it a powerful tool in Finsler geometry and we illustrate that by solving a number of named Fins- lerian 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 Berwaldian character of locally symmetric Finsler spaces, and extend the classic result of H.C. Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions. Our methods apply even in the absence of the strong convexity assumption usually as- sumed in Finsler geometry. The smoothness hypothesis can also be replaced to that of partial smoothness, a notion that we introduce in the paper. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.