Chern–Weil theory and the group of strict contactomorphisms

Abstract

In this paper we study the groups of contactomorphisms of a closed contact manifold from a topological viewpoint. First we construct examples of contact forms on spheres whose Reeb flow has a dense orbit. Then we show that the unitary group [Formula: see text] is homotopically essential in the group of contactomorphisms of the standard contact sphere [Formula: see text] and present a proof of the homotopy equivalence [Formula: see text]. In the second part of the paper we focus on the group of strict contactomorphisms — using the framework of Chern–Weil theory we introduce and study contact characteristic classes analogous to the Reznikov Hamiltonian classes in symplectic topology. We carry out several explicit calculations illustrating the nontriviality of these contact characteristic classes.