$\mathrm{GL}(2)$-structures in dimension four, $H$-flatness and integrability

Abstract

We show that torsion-free four-dimensional $\mathrm{GL}(2)$-structures are flat up to a coframe transformation with a mapping taking values in a certain subgroup $H\subset \mathrm{SL}(4,\mathbb{R})$ which is isomorphic to a semidirect product of the three-dimensional continuous Heisenberg group $H_3(\mathbb{R})$ and the Abelian group $\mathbb{R}$. In addition, we show that the relevant PDE system is integrable in the sense that it admits a dispersionless Lax-pair.

12 pages

Cited by in corpus (5)

• On deformations of the dispersionless Hirota equation
• Dispersionless integrable hierarchies and GL(2,R) geometry
• Multidimensional integrable systems from contact geometry
• GL(2)-geometry and complex structures
• The Cayley cubic and differential equations