The Binet–Legendre Metric in Finsler Geometry

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 the weaker notion of 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.

Completeness and incompleteness of the Binet–Legendre metric

“…In particular, the proofs of most results in the papers [24][25][26]35] could be based on any construction of Riemannian metric satisfying the above two conditions, at least when smooth and strictly convex Finsler metrics are considered. Several such constructions have been proposed in recent years.…”

“…We refer to [1,17,37] for discussions of further examples. The construction in [26] is called the Binet-Legendre metric 1 and has proven to be a flexible and useful tool in Finsler geometry, its definition will be recalled in Sect. 2.2.…”

Weakly conformal Finsler geometry

“…The Finsler metric is called the Zermelo transform of F with respect to W . This terminology has been first used by Matveev‐Troyanov in . The Zermelo transform of every Riemannian metric with respect to any appropriate vector field W is a Randers metric and vice versa; this is called the so‐called Zermelo correspondence in the contexts, cf.…”

“…Let us suppose that F is a Finsler metric on the manifold M. Given any vector field W ∈ X (M) satisfying [12]. The Zermelo transform of every Riemannian metric with respect to any appropriate vector field W is a Randers metric and vice versa; this is called the so-called Zermelo correspondence in the contexts, cf.…”

Sub-Finsler Structures from the Time-Optimal Control Viewpoint for some Nilpotent Distributions

“…The notion of sub-Finsler structure introduced above, when some non-smoothness of the norm is allowed, can be seen as a particular case of the following more general class: A partially smooth sub-Finsler structure on M is a triple (E, · E , f ) where E is a vector bundle over M , · E is a partially smooth Finsler structure on E (defined following Matveev and Troyanov [MT12]), and f : E → T M is a smooth morphism of bundles such that f (E p ) ⊆ T p M , for all p ∈ M . The norm on the induced distribution can be defined in analogy with (4), replacing w by (p, w) E .…”