15 Aug 2011; IMS (UiTø), room U1=A228. Monday 12:00-13:00.

Vladimir Matveev (Friedrich-Schiller-Universität Jena, Germany)

Partially smooth Finsler metrics and Binet-Legendre Riemannian metric

We introduce a new construction basing on the convex geometry that associates a Riemannian metric gF (called the Binet-Legendre metric) to a given Finsler metric F on a smooth manifold M. The transformation F → gF is C0-stable and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric gF also behave nicely under conformal or bi-Lipshitz deformation of the Finsler metric F that makes it a powerful tool in Finsler geometry.

We illustrate that by solving a number of named problems in Finsler geometry and giving short proofs of known results. In particular we answer a question of 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 and we solve a conjecture of Deng and Hu on locally symmetric Finsler spaces.

Our methods apply even in the absence of the strong convexity assumption usually assumed in Finsler geometry. The smoothness hypothesis can also be replaced to that partial smoothness, a notion that we introduce. Our results apply therefore to a vast class of Finsler metrics not usually considered in the Finsler literature.