Complex and Lagrangian Engel Structures

In this dissertation, 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 \emph{complex} Engel structure is an Engel 2-plane field on a complex surface for which the 2-planes are complex lines. A \emph{Lagrangian} Engel structure is an Engel 2-plane field on a symplectic 4-manifold for which the 2-planes are Lagrangian with respect to the symplectic structure. We solve the equivalence problems for complex Engel structures and Lagrangian Engel structures and use the resulting structure equations to classify homogeneous complex Engel structures and homogeneous Lagrangian Engel structures. This allows us to determine all compact, homogeneous examples.

For complex Engel structures, 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. For Lagrangian Engel structures, compact manifolds that support homogeneous Lagrangian Engel structures are diffeomorphic to quotients of one of a determined list of nilpotent or solvable 4-dimensional Lie groups by co-compact lattices.