Nonlinear generalized sections of vector bundles

We present an extension of J.F. Colombeau's theory of nonlinear generalized functions to spaces of generalized sections of vector bundles. Our construction builds on classical functional analytic notions, which is the key to having a canonical geometric embedding of vector bundle valued distributions into spaces of generalized sections. This permits to have tensor products, invariance under diffeomorphisms, covariant derivatives and the sheaf property. While retaining as much compatibility to L. Schwartz' theory of distributions as possible, our theory provides the basis for a rigorous and general treatment of singular pseudo-Riemannian geometry in the setting of Colombeau nonlinear generalized functions.