# Engel structures on complex surfaces

## TL;DR

This paper classifies complex surfaces that admit Engel structures as complex line bundles, showing this occurs precisely when the surfaces have trivial Chern classes, and constructs examples and associated geometric structures.

## Contribution

It provides a complete classification of complex surfaces with certain Engel structures and introduces new constructions and geometric decompositions.

## Key findings

- Surfaces with trivial Chern classes admit Engel structures as complex line bundles.
- Constructs explicit examples of such Engel structures.
- Defines a unique splitting of the tangent bundle related to the Engel structure.

## Abstract

We classify complex surfaces $(M,\,J)$ admitting Engel structures $\mathcal{D}$ which are complex line bundles. Namely we prove that this happens if and only if $(M,\,J)$ has trivial Chern classes. We construct examples of such Engel structures by adapting a construction due to Geiges. We also study associated Engel defining forms and define a unique splitting of $TM$ associated with $\mathcal{D}$ $J$-Engel.