We study the equation of flat connections in a given fiber bundle and discover a specific geometric structure on it, which we call a flat representation. We generalize this notion to arbitrary PDE and prove that flat representations of an equation are in 1–1 correspondence with morphisms , and being treated as submanifolds of infinite jet spaces. We show that flat representations include several known types of zero-curvature formulations of PDEs. In particular, the Lax pairs of the self-dual Yang–Mills equations and their reductions are of this type. With each flat representation ϕ we associate a complex Cϕ of vector-valued differential forms such that H1(Cϕ) describes infinitesimal deformations of the flat structure, which are responsible, in particular, for parameters in Bäcklund transformations. In addition, each higher infinitesimal symmetry S of defines a 1-cocycle cS of Cϕ. Symmetries with exact cS form a subalgebra reflecting some geometric properties of and ϕ. We show that the complex corresponding to itself is 0-acyclic and 1-acyclic (independently of the bundle topology), which means that higher symmetries of are exhausted by generalized gauge ones, and compute the bracket on 0-cochains induced by commutation of symmetries.
- Zero-curvature representations
- Self-dual Yang–Mills equations
- Nijenhuis bracket
- Flat connections
- Differential complexes