** The Binet-Legendre metric in Finsler geometry **

**1. Minkowski geometry **

I mean, geometry of a finite dimensional vectorspace equipped with a smooth, non-necessarily symmetric norm . Smooth means smooth away from the origin. Say is strictly convex if at each point (except possibly at the origin), is positive definite. In this case, is a Riemannian metric on the complement of the origin.

By homogeneity, for all ,

**2. Finsler geometry **

A *Finsler metric* on a smooth manifold is a continuous function which is a non-necessarily symmetric norm on each tangent space . Equivalently, it is a 1-homogeneous Lagrangian.

The length of curves iswell-defined, and distance as well. It is a non-reversible distance.

Theorem 1 (Ivanov 2009)If is a reversible geodesic distance on a manifold which is locally bi-Lipschitz equivalent to some Riemannian metric, then is weakly Finslerian, i.e. defined by a Borel Finsler metric.

We are interested in the case when is smooth, and strictly convex on fibers. The fundamental tensor is a Euclidean structure on the vectorbundle over zero section. From it, one can define a connection (known as Chern connection), a notion of curvature (known as flag curvature) which gives rise to comparison theorems. For instance, a complete simply connected Finsler manifold with nonpositive flag curvature is . A complete simply connected Finsler manifold with -pinched flag curvature is homeomorphic to a sphere.

**3. Relating a Finsler metric to Riemannian metrics **

There are many ways to canonically associate an ellipsoid to a convex set.

** 3.1. Choices **

The John ellipsoid is the ellipsoid containing of minimal volume. It is uniquely defined, but nevertheless need not depend differentiably on .

The Binet ellipsoid is defined in the dual space . It is the unit ball of the following quadratice form

In other words, it computes the moments of inertia of .

The Binet-Legendre ellipsoid is defined by the dual quadratic form on . This provides us with a Riemannian metric associated to a Finsler metric . There is no loss of differentiability. If is Riemannian, .

** 3.2. Consequences **

1. The isometry group of a Finsler manifold is a Lie group.

2. If a Finsler manifold admits an essential conformal mapping, i.e. a conformal mapping which is not an isometry of a conformally equivalent Finsler metric, then

- either Finsler manifold is isometric to a Minkowski space,
- or Finsler manifold is conformally equivalent to the round Riemannian sphere.

The bulk of the proof lies in Riemannian geometry, work by many people (Obata, Ferrand, Alekseevski,…).

**4. Questions **

Are there materials which are sufficiently anisotropic that this becomes useful ? Motion in presence of a force depending on position or on position and speed can be modelled by a Finsler metric (e.g. Zermelo navigation problem).

Are geodesics specified by initial point and speed ? Yes if is strictly convex.