Riemannian Properties of Engel Structures

This paper is about geometric and Riemannian properties of Engel structures. A choice of defining forms for an Engel structure D determines a distribution R transverse to D called the Reeb distribution. We study conditions that ensure integrability of R. For example, if we have a metric g that makes the splitting TM= D\oplus R orthogonal and such that 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.