Affine connections on complex compact surfaces and Riccati distributions

TL;DR

This paper establishes a correspondence between torsion-free affine connections on complex surfaces and Riccati distributions, linking geometric structures with foliations, especially in the compact case where affine structures correspond to Riccati foliations.

Contribution

It introduces a novel correspondence between affine connections and Riccati distributions on complex surfaces, extending to Riccati foliations in the compact case.

Findings

One-to-one correspondence between affine connections and Riccati distributions.

Extension of the correspondence to Riccati foliations on compact surfaces.

Provides a new geometric framework linking affine structures and foliations.

Abstract

Let $M$ be a complex surface. We show that there is a one-to-one correspondence between torsion-free affine connections on $M$ and Riccati distributions on $\mathbb{P}(TM)$. Furthermore, if $M$ is compact, then this correspondence induces a one-to-one correspondence between affine structures on $M$ and Riccati foliations on $\mathbb{P}(TM)$.