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 D'(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 D'(M) into Ĝ(M) that renders C∞(M) a faithful subalgebra of Ĝ(M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus Ĝ(M) retains all the distinguishing properties of the local theory in a global context. © 2002 Elsevier Science (USA).