Colombeau algebras on a \(C^{\infty{}}\)-manifold
The notion of a distribution in the sense of Colombeau is extended to an arbitrary \(C^ \infty\)-manifold \(X\). The set \({\mathcal G}\) of all such distributions is defined as a quotient sheaf \({\mathcal M}/{\mathcal N}\) of algebras \({\mathcal M}\) and \({\mathcal N}\). The existence of an injective sheaf morphism \(\sigma:{\mathcal D}'\to{\mathcal G}\) on \(X\) is proved such that for any section \(u\in{\mathcal D}'(\Omega)\), \(\Omega\subset X\), \(\sigma(u)\sim u\) holds, and that \(\sigma\) is the identity on the subsheaves \(C^ \infty\) of \(C^ \infty\)-functions in \({\mathcal D}'\) and \({\mathcal G}\). A theorem for submanifolds \(Y\subset X\) is also provided.
scientific article; zbMATH DE number 57316