Grassmannians Gr(N − 1, N + 1), closed differential N − 1-forms and N-dimensional integrable systems**In memory of S V Manakov.

Abstract

Integrable flows on the Grassmannians Gr(N − 1, N + 1) are defined by the requirement of closedness of the differential N − 1-forms ΩN − 1 of rank N − 1 naturally associated with Gr(N − 1, N + 1). Gauge-invariant parts of these flows, given by the systems of the N − 1 quasi-linear differential equations, describe coisotropic deformations of (N − 1)-dimensional linear subspaces. For the class of solutions which are Laurent polynomials in one variable these systems coincide with N-dimensional integrable systems such as the Liouville equation (N = 2), dispersionless Kadomtsev–Petviashvili equation (N = 3), dispersionless Toda equation (N = 3), Plebanski second heavenly equation (N = 4) and others. Gauge-invariant part of the forms ΩN − 1 provides us with the compact form of the corresponding hierarchies. Dual quasi-linear systems associated with the projectively dual Grassmannians Gr(2, N + 1) are defined via the requirement of the closedness of the dual forms Ω⋆N − 1. It is shown that at N = 3 the self-dual quasi-linear system, which is associated with the harmonic (closed and co-closed) form Ω2, coincides with the Maxwell equations for orthogonal electric and magnetic fields.