Communications in Analysis and Geometry

Volume 27 (2019)

Number 8

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

Pages: 1851 – 1868



Wojciech Kryński (Institute of Mathematics, Polish Academy of Sciences, Warsaw, Poland)

Thomas Mettler (Institut für Mathematik, Goethe-Universität Frankfurt, Frankfurt am Main, Germany)


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.

Received 29 November 2016

Accepted 15 September 2017

Published 21 January 2020