The regularity and products in contact geometry

TL;DR

This paper characterizes regular contact manifolds with complete Reeb vector fields as principal $S^1$ or $R$-bundles, extending the Boothby-Wang theorem without assuming compactness, and introduces contact products related to prequantization bundles.

Contribution

It provides a general proof that regular contact manifolds are principal bundles with specific structure groups, removing the compactness assumption, and defines contact products for manifolds with commensurate Reeb periods.

Findings

Regular contact manifolds are principal $S^1$ or $R$-bundles.

The proof extends the Boothby-Wang theorem to non-compact cases.

Contact products exist when Reeb periods are commensurate.

Abstract

We study regular contact manifolds $(M,\eta)$ whose Reeb vector field is complete and prove that they are canonically principal bundles with the structure group $S^1$ or $\mathbb{R}$. For compact $M$, our proof is very short and elementary and covers the celebrated Boothby-Wang theorem, but we do not assume compactness from the very beginning. However, to prove our result in full generality we use some topological tools adapted to smooth fibrations. In the second part of the paper, we describe a natural concept of contact products of general contact manifolds as well as a product of principal contact manifolds, which exists if the periods of the Reeb vector fields are commensurate, and corresponds to the construction of products of prequantization bundles of symplectic manifolds.