A Global Theory of Algebras of Generalized Functions

Abstract

We present a geometric approach to defining an algebra Ĝ (M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space DOE(M) of distributions on M. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of Ĝ (M). Ĝ (M) is a differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of

Generalized pseudo-Riemannian geometry

“…Recently the theory of algebras of generalized functions (Colombeau algebras) [7], [8] has been restructured to allow for applications in a geometrical context [12,13,24]. The need for the latter has been clearly demonstrated by the use of nonlinear generalized function methods in the field of Lie group analysis of partial differential equations (e.g., [21], [9]) and the study of singular spacetimes in general relativity (see [40] for a survey).…”

Remarks on the distributional Schwarzschild geometry

“…Note that, in general, a regularization procedure depends on the coordinate system it is performed in (for a diffeomorphism invariant notion of regularization using paths of mollifiers see [16,17]). However, since we had to fix the differentiable structure beforehand this is no further restriction.…”

Isomorphisms of algebras of generalized functions

“…Under the influence of applications of a more geometric nature (e.g. in Lie group analysis of differential equations and in general relativity), a geometric theory of Colombeau generalized functions arose [6,7,10]. In particular, for X, Y smooth paracompact Hausdorff manifolds, Colombeau generalized functions from X to Y can be defined.…”