Nonlinear generalized sections of vector bundles

TL;DR

This paper extends Colombeau's nonlinear generalized functions to vector bundle sections, enabling geometric operations and a rigorous framework for singular pseudo-Riemannian geometry.

Contribution

It introduces a canonical geometric embedding of vector bundle valued distributions into generalized sections, facilitating tensor operations and invariance under diffeomorphisms.

Findings

Provides a canonical embedding of distributions into generalized sections

Enables tensor products and covariant derivatives in the generalized setting

Supports a rigorous approach to singular pseudo-Riemannian geometry

Abstract

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.