Abstract:

We introduce a construction that associates a Riemannian metric  (called the metric) to a given Finsler metric on a smooth manifold . The transformation  is and has good smoothness properties, in contrast to previously considered constructions. The Riemannian metric also behaves nicely under conformal or isometric transformations of the Finsler metric that makes it a powerful tool in Finsler geometry. We illustrate that by solving a number of named problems in Finsler geometry. In particular we extend a classical result of Wang to all dimensions. We answer a question of Matsumoto about local conformal mapping between two Berwaldian spaces and use it to investigation of essentially conformally Berwaldian manifolds. We describe all possible conformal self maps and all self similarities on a Finsler manifold, generasing the famous result of Obata to Finslerian manifolds. We also classify all compact conformally flat Finsler manifolds. We solve a conjecture of Deng and Hou on locally symmetric Finsler spaces. We prove smoothness of isometries of Holder-continuous Finsler metrics. We construct new ``easy to calculate'' conformal and metric invariants of finsler manifolds. The results are based on the papers arXiv:1104.1647, arXiv:1409.5611, arXiv:1408.6401, arXiv:1506.08935, arXiv:1406.2924 partially joint with M. Troyanov (EPF Lausanne) and Yu. Nikolayevsky (Melbourne)