GL(2)‐geometry and complex structures

We study GL(2)‐structures on differential manifolds. A GL(2)‐structure is a smooth field of rational normal curves in the tangent bundle of a manifold. We provide an explicit construction of a canonical connection for any GL(2)‐structure in any dimension greater than 3. Moreover, we prove that any GL(2)‐structure on an even‐dimensional manifold defines a certain almost‐complex structure on a bundle over the original manifold. Further, we exploit the almost‐complex structure to study various notions of integrability of GL(2)‐structures. For this, we develop a twistor‐like construction. In particular, we prove that in dimension 4 a GL(2)‐structure is torsion‐free if and only if the associated almost‐complex structure is integrable.