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 (15)

• Differential equations and conformal structures
• Paraconformal geometry of $n$th order ODEs, and exotic holonomy in dimension four
• On the Einstein-Weyl and conformal self-duality equations
• Differential Geometry from Differential Equations
• Einstein--Weyl geometry, dispersionless Hirota equation and Veronese webs
• Integrable Background Geometries
• Veronese webs and nonlinear PDEs
• On Kähler metrisability of two-dimensional complex projective structures
• Weyl metrisability of two-dimensional projective structures
• Differential Geometric Aspects of Causal Structures
• Reduction of beta-integrable 2-Segre structures
• $\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability
• Dispersionless integrable hierarchies and GL(2,R) geometry
• The Cayley cubic and differential equations
• Causal geometries and third-order ordinary 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