Conformal vector fields and Finsler Geometry

Marc Troyanov, 27th November 2008, MA 12

A Finsler metric on a manifold M is given by a  function F: TM - > R on the tangent bundle which defines a norm on each tangent space. It is assumed that F is smooth on the complement of the zero section. Finsler metrics are thus a generalization of Riemannian metrics. Some Riemannian concepts  do extend to the Finsler case and some don't. The concepts of isometry, conformal map,  Killing  fields and conformal fields do extend to the Finsler case. In this talk, we will classify all Finsler conformal fields. We will show in particular that if M is not diffeomorphic to  R^n
or S^n, then any conformal field is essentially a Killing field (Lichnerowicz conjecture).