2017/05/18 by Doubrov, Boris, Ferapontov, Eugene, Kruglikov, Boris +1
#Differential Geometry (math.DG) #Exactly Solvable and Integrable Systems (nlin.SI) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph)
paper · doi:10.48550/arxiv.1705.06999
Let Gr(d,n) be the Grassmannian of d-dimensional linear subspaces of an n-dimensional vector space V. A submanifold X⊂ Gr(d, n) gives rise to a differential system Σ(X) that governs d-dimensional submanifolds of V whose Gaussian image is contained in X. We investigate a special case of this construction where X is a sixfold in Gr(4, 6). The corresponding system Σ(X) reduces to a pair of first-order PDEs for 2 functions of 4 independent variables. Equations of this type arise in self-dual Ricci-flat geometry. Our main result is a complete description of integrable systems Σ(X). These naturally fall into two subclasses. (1) Systems of Monge-Ampère type. The corresponding sixfolds X are codimension 2 linear sections of the Plücker embedding Gr(4,6)⊂ℙ14. (2) General linearly degenerate systems. The corresponding sixfolds X are the images of quadratic maps ℙ6- Gr(4, 6) given by a version of the classical construction of Chasles. We prove that integrability is equivalent to the requirement that the characteristic variety of system Σ(X) gives rise to a conformal structure which is self-dual on every solution. In fact, all solutions carry hyper-Hermitian geometry.