Foundations of a Nonlinear Distributional Geometry

Colombeau’s construction of generalized functions (in its special variant) is extended to a theory of generalized sections of vector bundles. As particular cases, generalized tensor analysis and exterior algebra are studied. A point value characterization for generalized functions on manifolds is derived, several algebraic characterizations of spaces of generalized sections are established and consistency properties with respect to linear distributional geometry are derived. An application to nonsmooth mechanics indicates the additional flexibility offered by this approach compared to the purely distributional picture. Mathematics Subject Classification (2000): Primary: 46F30; secondary: 46T30, 46F10, 37J05