Integrability of Second-Order Partial Differential Equations and the Geometry of GL(2)-Structures

Abstract

A GL(2,R)-structure on a smooth manifold of dimension n+1 corresponds to a distribution of non-degenerate rational normal cones over the manifold. Such a structure is called k-integrable if there exist many foliations by submanifolds of dimension k whose tangent spaces are spanned by vectors in the cones.This structure was first studied by Bryant for n=3 and k=2. The work included here (n=4 and k=2,3) was suggested by Ferapontov, et al., who showed that the cases (n=4,k=2) and (n=4, k=3) can arise from integrability of second-order PDEs via hydrodynamic reductions.Cartan--Kahler analysis for n=4 and k=3 leads to a complete classification of local structures into 54 equivalence classes determined by the value of an essential 9-dimensional representation of torsion for the GL(2,R)-structure. These classes are described by the factorization root-types of real binary octic polynomials. Each of these classes must arise from a PDE, but the PDEs remain to be identified. Also, we study the local problem for n >= 5 and k=2,3 and conjecture that similar classifications exist for these cases; however, the interesting integrability results are essentially unique to degree 4. The approach is that of moving frames, using Cartan's method of equivalence, the Cartan--Kahler theorem, and Cartan's structure theorem.