Tight contact structures and Anosov flows

Abstract

In this review, we demonstrate how classic and contemporary results on the classification of tight contact structures apply to the problem of existence and uniqueness of Anosov flows on three-manifolds. The ingredients we use are the results of Mitsumatsu on Anosov flows, the homotopy invariant of plane fields as described by Gompf and others, and certain recent classification results of Giroux and Honda. A simple example is a novel proof of the nonexistence of Anosov flows on S 3 using only contact topology (and in particular without use of Novikov's Theorem on foliations).