Foundations of a nonlinear distributional geometry

TL;DR

This paper develops a nonlinear distributional geometry framework by extending Colombeau's generalized functions to sections of vector bundles, enabling advanced tensor analysis and applications in nonsmooth mechanics.

Contribution

It introduces a generalized sections theory extending Colombeau's functions, with algebraic characterizations and consistency with linear distributional geometry, enhancing nonsmooth analysis.

Findings

Established a point value characterization for generalized functions on manifolds.

Developed algebraic characterizations of generalized sections.

Applied the framework to nonsmooth mechanics, demonstrating increased flexibility.

Abstract

Co lombeau'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.