On the integrability in Grassmann geometries: integrable systems associated with fourfolds Gr(3, 5)

Abstract

We investigate dispersionless integrable systems in 3D associated with fourfolds in the Grassmannian Gr(3,5). Such systems appear in numerous applications in continuum mechanics, general relativity and differential geometry, and include such well-known examples as the dispersionless Kadomtsev-Petviashvili equation, the Boyer-Finley equation, etc. We prove the equivalence of the four different approaches to integrability, revealing a remarkable correspondence with Einstein-Weyl geometry and the theory of GL(2,R) structures.

This is an elaborated version of the main part concerning dispersionless integrable systems in 3D, reflected in the title of the paper. We omitted the last two sections on systems in 4D and higher dimensional Monge-Ampere equations that will be expanded and posted in arXiv separately in the near future. These parts, as well as supplementary materials, are still accessible via arXiv:1503.02274v1