Complex Engel Structures
Abstract
We study the geometry of Engel structures, which are 2plane fields on 4manifolds satisfying a generic condition, that are compatible with other geometric structures. A complex Engel structure is an Engel 2plane field on a complex surface for which the 2planes 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 cocompact lattices, where $G$ is the connected and simplyconnected Lie group with Lie algebra $\mathfrak{sl}_2(\mathbb{R})$ and $H$ is a solvable Lie group.
 Publication:

arXiv eprints
 Pub Date:
 May 2018
 arXiv:
 arXiv:1805.07660
 Bibcode:
 2018arXiv180507660Z
 Keywords:

 Mathematics  Differential Geometry