Chern-Weil theory and the group of strict contactomorphisms

TL;DR

This paper explores the topology of contactomorphism groups on closed contact manifolds, constructing examples, analyzing homotopy types, and introducing contact characteristic classes via Chern-Weil theory.

Contribution

It introduces contact characteristic classes using Chern-Weil theory and provides explicit calculations demonstrating their non-triviality, advancing understanding of contactomorphism groups.

Findings

U(n+1) is homotopically essential in the contactomorphism group of the standard sphere

The contactomorphism group of the 3-sphere is homotopy equivalent to U(2)

Contact characteristic classes are non-trivial, as shown by explicit calculations

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 U(n+1) is homotopically essential in the group of contactomorphisms of the standard contact sphere S^(2n+1) and prove that in the case of the 3-sphere the contactomorphism group is in fact homotopy equivalent to U(2). 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 non-triviality of the contact classes.