1999/12/28 by Michael Grosser, Michael Kunzinger, Roland Steinbauer +1
Mathematics · Physics and Astronomy · #math.FA #math-ph #math.MP #msc:46F30 #msc:46T30
published as Advances in Math. 166 (2002) 50-72 · 24 pages, LaTeX
arxiv created 1999/12/28 · arxiv updated 2009/11/30
We present a geometric approach to defining an algebra \mathcal G(M) (the Colombeau algebra) of generalized functions on a smooth manifold M containing the space \mathcal D'(M) of distributions on M. Based on differential calculus in convenient vector spaces we achieve an intrinsic construction of \mathcal G(M). \mathcal G(M) is a\em differential algebra, its elements possessing Lie derivatives with respect to arbitrary smooth vector fields. Moreover, we construct a canonical linear embedding of \mathcal D'(M) into \mathcal G(M) that renders \mathcal C^∞ (M) a faithful subalgebra of \mathcal G(M). Finally, it is shown that this embedding commutes with Lie derivatives. Thus \mathcal G(M) retains all the distinguishing properties of the local theory in a global context.