Regular contact manifolds: a generalization of the Boothby-Wang theorem

TL;DR

This paper extends the Boothby-Wang theorem to regular contact manifolds with complete Reeb vector fields, showing they form principal bundles over symplectic manifolds with a compatible prequantization structure, without requiring compactness.

Contribution

It generalizes the Boothby-Wang theorem by characterizing regular contact manifolds with complete Reeb vector fields as principal bundles over symplectic manifolds, including non-compact cases.

Findings

The fibration is either an $S^1$- or an $ extbf{R}$-principal bundle.

Existence of a unique symplectic form on the orbit space compatible with the contact form.

The symplectic form admits a prequantization in the $S^1$-bundle case.

Abstract

A regular contact manifold is a manifold $M$ equipped with a globally defined contact form $\eta$ such that the topological space $M/\mathcal{R}$ of orbits (trajectories) of the Reeb vector field $\mathcal{R}$ of $\eta$ carries a smooth manifold structure, so the canonical projection $p:M\to M/\mathcal{R}$ is a smooth fibration. We show that, under the additional assumption that $\mathcal{R}$ is a complete vector field, this fibration is actually either an $S^1$- or an $\mathbb{R}$-principal bundle. Moreover, there exists a unique symplectic form $\omega$ on $M/\mathcal{R}$ such that $p^*(\omega)=\mathrm{d}\eta$ which is $\rho$-integral in the $S^1$-bundle case, where $\rho$ is the minimal period of the $S^1$-action, so the symplectic manifold $(M/\mathcal{R},\omega)$ admits a prequantization. We do not assume that $M$ is compact.