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

TL;DR

This paper explores dispersionless integrable systems linked to fourfolds in Grassmannian Gr(3,5), revealing their equivalence through multiple approaches and connections to Einstein-Weyl geometry and GL(2,R) structures.

Contribution

It demonstrates the equivalence of four different integrability approaches and uncovers their deep geometric connections in Grassmannian contexts.

Findings

Equivalence of four integrability approaches

Connection to Einstein-Weyl geometry

Relation to GL(2,R) structures

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.