## Notes of Marc Troyanov’s Cambridge lecture 14-03-2017

The Binet-Legendre metric in Finsler geometry

1. Minkowski geometry

I mean, geometry of a finite dimensional vectorspace ${V}$ equipped with a smooth, non-necessarily symmetric norm ${F}$. Smooth means smooth away from the origin. Say ${F}$ is strictly convex if at each point (except possibly at the origin), ${Hess(F^2)}$ is positive definite. In this case, ${h=\frac{1}{2}Hess(F^2)}$ is a Riemannian metric on the complement of the origin.

By homogeneity, for all ${v\in V\setminus\{0\}}$,

$\displaystyle \begin{array}{rcl} F(v)^2=h_v(v,v). \end{array}$

2. Finsler geometry

A Finsler metric on a smooth manifold ${M}$ is a continuous function ${F:TM\rightarrow{\mathbb R}}$ which is a non-necessarily symmetric norm on each tangent space ${T_p M}$. Equivalently, it is a 1-homogeneous Lagrangian.

The length of ${C^1}$ curves ${\int F(\gamma(t),\dot\gamma(t))\,dt}$ iswell-defined, and distance as well. It is a non-reversible distance.

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

We are interested in the case when ${F}$ is smooth, and strictly convex on fibers. The fundamental tensor ${h}$ is a Euclidean structure on the vectorbundle ${\pi^*TM}$ over ${T^0 M=TM\setminus}$ 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 ${CAT(0)}$. A complete simply connected Finsler manifold with ${\frac{1}{4}}$-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 ${J(K)}$ is the ellipsoid containing ${K}$ of minimal volume. It is uniquely defined, but nevertheless need not depend differentiably on ${K}$.

The Binet ellipsoid is defined in the dual space ${V^*}$. It is the unit ball of the following quadratice form

$\displaystyle \begin{array}{rcl} g^*(\theta,\phi)=\frac{n+2}{vol(K)}\int_{K}\theta(v)\phi(v)\,dv \end{array}$

In other words, it computes the moments of inertia of ${K}$.

The Binet-Legendre ellipsoid is defined by the dual quadratic form on ${V}$. This provides us with a Riemannian metric ${g_F}$ associated to a Finsler metric ${F}$. There is no loss of differentiability. If ${F}$ is Riemannian, ${g_F=\sqrt{F}}$.

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 ${F}$ is strictly convex.