A global theory of algebras of generalizedfunctionsM

We present a geometric approach to deening an algebra ^ G(M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space D 0 (M) of distributions on M. Based on diierential calculus in convenient vector spaces we achieve an intrinsic construction of ^ G(M). ^ G(M) is a diierential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector elds. Moreover, we construct a canonical linear embedding of D 0 (M) into ^ G(M) that renders C 1 (M) a faithful subalgebra of ^ G(M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus ^ G(M) retains all the distinguishing properties of the local theory in a global context.