To every Finsler norm \(F\), the Binet-Legendre (Riemannian) metric \(g_F\) is associated, which has remarkable natural functorial properties. The transformation \(F \mapsto g_F\) is \(C^0\)-stable and has good smoothness properties, in contrast to previous alternative constructions of this kind. The Riemannian metric \(g_F\) also behaves nicely under conformal or bi-Lipschitz deformation of the Finsler metric \(F\). These properties transform this metric into a powerful tool for solving several Finslerian geometric problems, and for generalizing or providing new and shorter proofs to known results.
In particular, the authors answer a question of M. Matsumoto about local conformal mappings between two Minkowski spaces, and describe all possible conformal self-maps and all self-similarities on a Finsler manifold. As well, further results refer to the classification of all compact conformally flat Finsler manifolds, the solving of a conjecture of S. Deng and Z. Hou on the Berwaldian character of locally symmetric Finsler spaces, and the extension of the classic result of H. C. Wang about the maximal dimension of the isometry groups of Finsler manifolds to manifolds of all dimensions.
It is notable that the employed methods apply even in the absence of the strong convexity assumption – usually assumed in Finsler geometry – and that the smoothness hypothesis can also be replaced by the notion of “partial smoothness” (defined by the authors). Consequently, the obtained results apply to a vast class of Finsler metrics, which are not usually considered in the Finsler literature.