Contact structures on principal circle bundles

Abstract

We describe a necessary and sufficient condition for a principal circle bundle over an even-dimensional manifold to carry an invariant contact structure. As a corollary, it is shown that if the trivial circle bundle over a given base manifold carries an invariant contact structure, then so do all circle bundles over that base. In particular, all circle bundles over 4-manifolds admit invariant contact structures. We also discuss the Bourgeois construction of contact structures on odd-dimensional tori in this context, and we relate our results to recent work of Massot, Niederkrüger and Wendl on weak symplectic fillings in higher dimensions.