Abstract Differential Geometry, Differential Algebras of Generalized Functions, and de Rham Cohomology

Abstract

Abstract differential geometry is a recent extension of classical differential geometry on smooth manifolds which, however, does no longer use any notion of Calculus. Instead of smooth functions, one starts with a sheaf of algebras, i.e., the structure sheaf, considered on an arbitrary topological space, which is the base space of all the sheaves subsequently involved. Further, one deals with a sequence of sheaves of modules, interrelated with appropriate ‘differentials’, i.e., suitable ‘Leibniz’ sheaf morphisms, which will constitute the ‘differential complex’. This abstract approach captures much of the essence of classical differential geometry, since it places a powerful apparatus at our disposal which can reproduce and, therefore, extend fundamental classical results. The aim of this paper is to give an indication of the extent to which this apparatus can go beyond the classical framework by including the largest class of singularities dealt with so far. Thus, it is shown that, instead of the classical structure sheaf of algebras of smooth functions, one can start with a significantly larger, and nonsmooth, sheaf of so-called nowhere dense differential algebras of generalized functions. These latter algebras, which contain the Schwartz distributions, also provide global solutions for arbitrary analytic nonlinear PDEs. Moreover, unlike the distributions, and as a matter of physical interest, these algebras can deal with the vastly larger class of singularities which are concentrated on arbitrary closed, nowhere dense subsets and, hence, can have an arbitrary large positive Lebesgue measure. Within the abstract differential geometric context, it is shown that, starting with these nowhere dense differential algebras as a structure sheaf, one can recapture the exactness of the corresponding de Rham complex, and also obtain the short exponential sequence. These results are the two fundamental ingredients in developing differential geometry along classical, as well as abstract lines. Although the commutative framework is used here, one can easily deal with a class of singularities which is far larger than any other one dealt with so far, including in noncommutative theories.