Riemannian Properties of Engel Structures

Abstract

Abstract This paper is about geometric and Riemannian properties of Engel structures. A choice of defining forms for an Engel structure $\mathcal{D}$ determines a distribution $\mathcal{R}$ transverse to $\mathcal{D}$ called the Reeb distribution. We study conditions that ensure integrability of $\mathcal{R}$. For example, if we have a metric $g$ that makes the splitting $TM=\mathcal{D}\oplus \mathcal{R}$ orthogonal and such that $\mathcal{D}$ is totally geodesic then there exists another Reeb distribution, which is integrable. We introduce the notion of K-Engel structures in analogy with K-contact structures, and we classify the topology of K-Engel manifolds. As natural consequences of these methods, we provide a construction that is the analogue of the Boothby–Wang construction in the contact setting, and we give a notion of contact filling for an Engel structure.