The Berger conjecture for three dimensional manifolds

Abstract

A conjecture of Berger says that on a simply connected manifold all of whose geodesics are closed, all unit speed geodesics have the same shortest period. In this paper we will show that the S^1-action induced by the geodesic flow is free. We use symplectic geometry to investigate the topology of the quotient of the unit tangent bundle T^1M/S^1 by the S^1 action that comes from the geodesic flow. We show that this quotient admits the structure of a smooth symplectic manifold and that this symplectic manifold is symplectomorphic to S^2 \times S^2 equipped with a split form. This will enable a computation involving the Euler class of the orbibundle T^1M\rightarrow T^1M/S^1 which will lead to a solution of Berger's conjecture for three-dimensional manifolds.