Smith, Abraham D. Integrable \(\mathrm{GL}(2)\) geometry and hydrodynamic partial differential equations. (English) Zbl 1239.58020 Commun. Anal. Geom. 18, No. 4, 743-790 (2010). From the author’s abstract: “This article is a local analysis of integrable \(\mathrm{GL}(2)\)-structures of degree 4. A \(\mathrm{GL}(2)\)-structure of degree n corresponds to a distribution of rational normal cones over a manifold \(M\) of dimension \((n+1)\). Integrability corresponds to the existence of many submanifolds that are spanned by lines in the cones. These \(\mathrm{GL}(2)\)-structures are important because they naturally arise from a certain family of second-order hyperbolic PDEs in three variables that are integrable via hydrodynamic reduction. Familiar examples include the wave equation, the first flow of the dKP equation, and the Boyer-Finley equation. The main results are a structure theorem for integrable \(\mathrm{GL}(2)\)-structures, a classification for connected integrable \(\mathrm{GL}(2)\)-structures, and an equivalence between local integrable \(\mathrm{GL}(2)\)-structures and Hessian hydrodynamic hyperbolic PDEs in three variables. This yields natural geometric characterizations of the wave equation, the first flow of the dKP hierarchy, and several others. It also provides an intrinsic, coordinate-free infrastructure to describe a large class of hydrodynamic integrable systems in three variables.” Reviewer: Michael Kunzinger (Wien) Cited in 4 Documents MSC: 58J45 Hyperbolic equations on manifolds 58A15 Exterior differential systems (Cartan theory) 37K10 Completely integrable infinite-dimensional Hamiltonian and Lagrangian systems, integration methods, integrability tests, integrable hierarchies (KdV, KP, Toda, etc.) 35Q35 PDEs in connection with fluid mechanics 53C10 \(G\)-structures 58A20 Jets in global analysis 35A30 Geometric theory, characteristics, transformations in context of PDEs 35L20 Initial-boundary value problems for second-order hyperbolic equations 35L05 Wave equation Keywords:wave equation; structure theorem; integrable \(\mathrm{GL}(2)\)-structures; classification; equivalence; hydrodynamic integrable systems PDFBibTeX XMLCite \textit{A. D. Smith}, Commun. Anal. Geom. 18, No. 4, 743--790 (2010; Zbl 1239.58020) Full Text: DOI arXiv