SCALAR DIFFERENTIAL INVARIANTS, DIFFIETIES AND CHARACTERISTIC CLASSES

Abstract

This chapter provides a history of the theory of scalar differential invariants. Spectra of scalar differential invariants algebras are diffieties. By diffieties, objects of the category of differential equations are denoted. Diffieties are a kind of manifolds, generally infinite-dimensional, supplied with a finite-dimensional distribution called the Cartan distribution or the infinite order-contact structure. The chapter describes some geometrical structures and their scalar differential invariants and characteristic classes. It further discusses a few generalizations and variants of the preceding considerations and related problems. The chapter also discusses the topics of natural equations and special characteristic classes, characteristic classes associated with Lie pseudo-groups, Tsujishita's approach and the secondary quantizied-characteristic classes, quasi-homogeneous geometries, and nonhomogeneous geometrical structures and singularities.