GL(2)-geometry and complex structures

TL;DR

This paper explores $GL(2)$-structures on manifolds, revealing their connection to almost-complex structures, integrability conditions, and twistor constructions, with applications to differential equations and geometric theory.

Contribution

It introduces a new link between $GL(2)$-structures and almost-complex structures, along with a canonical connection and integrability criteria, advancing geometric understanding.

Findings

$GL(2)$-structures induce almost-complex structures on bundles.

Integrability of $GL(2)$-structures relates to almost-complex structure integrability.

A canonical connection for $GL(2)$-structures is explicitly constructed.

Abstract

We study $GL(2)$-structures on differential manifolds. The structures play a fundamental role in the geometric theory of ordinary differential equations. We prove that any $GL(2)$-structure on an even dimensional manifold give rise to a certain almost-complex structure on a bundle over the original manifold. Further, we exploit a natural notion of integrability for the $GL(2)$-structures, which is a counterpart of the self-duality for the 4-dimensional conformal structures. We relate the integrability of the $GL(2)$-structures to the integrability of the almost-complex structures. This allows to perform a twistor-like construction for the $GL(2)$-geometry. Moreover, we provide an explicit construction of a canonical connection for any $GL(2)$-structure.