Complex Engel Structures

We study the geometry of Engel structures, which are 2-plane fields on 4-manifolds satisfying a generic condition, that are compatible with other geometric structures. A complex Engel structure is an Engel 2-plane field on a complex surface for which the 2-planes are complex lines. We solve the equivalence problems for complex Engel structures and use the resulting structure equations to classify homogeneous complex Engel structures. This allows us to determine all compact, homogeneous examples. Compact manifolds that support homogeneous complex Engel structures are diffeomorphic to $S^1\times SU(2)$ or quotients of $\mathbb{C}^2$, $S^1\times SU(2)$, $S^1\times G$ or $H$ by co-compact lattices, where $G$ is the connected and simply-connected Lie group with Lie algebra $\mathfrak{sl}_2(\mathbb{R})$ and $H$ is a solvable Lie group.