Nonlinear generalised functions on manifolds

TL;DR

This paper develops a global nonlinear theory of differential geometry on manifolds using Colombeau algebras and smoothing operators, enabling advanced analysis of tensor distributions.

Contribution

It introduces a new approach to construct algebras of generalized functions on manifolds, suitable for differential geometry applications, and defines a generalized Lie derivative.

Findings

Generalized Lie derivative commutes with distribution embedding.

Covariant derivative of generalized scalar fields commutes with embedding.

Framework is suitable for future applications in nonlinear differential geometry.

Abstract

This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper which is based on Colombeau algebras of tensor distributions on manifolds. We adopt a new approach and construct a global theory of algebras of generalised functions on manifolds based on the concept of smoothing operators. This produces a generalisation of previous theories in a form which is suitable for applications to differential geometry. The generalised Lie derivative is introduced and shown to commute with the embedding of distributions. It is also shown that the covariant derivative of a generalised scalar field commutes with this embedding at the level of association.