GL(2)-geometry and complex structures

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.

References in corpus (2)

• Integrable Background Geometries
• The Cayley cubic and differential equations

Cited by in corpus (2)

• Weyl metrizability of 3-dimensional projective structures and CR submanifolds
• Holomorphic GL(2)-geometry on compact complex manifolds