A global theory of algebras of generalized functions II: tensor distributions

TL;DR

This paper develops a comprehensive algebraic framework for generalized tensor fields on smooth manifolds, enabling consistent embedding of tensor distributions and applications in nonlinear geometry and low-regularity spacetimes.

Contribution

It introduces a universal algebra of generalized tensor fields with optimal embedding and compatibility properties, extending previous constructions to tensor distributions.

Findings

Provides a canonical embedding of tensor distributions into the algebra

Ensures compatibility with Lie derivatives for tensor fields

Facilitates applications in nonlinear distributional geometry and general relativity

Abstract

We extend the construction of [19] by introducing spaces of generalized tensor fields on smooth manifolds that possess optimal embedding and consistency properties with spaces of tensor distributions in the sense of L. Schwartz. We thereby obtain a universal algebra of generalized tensor fields canonically containing the space of distributional tensor fields. The canonical embedding of distributional tensor fields also commutes with the Lie derivative. This construction provides the basis for applications of algebras of generalized functions in nonlinear distributional geometry and, in particular, to the study of spacetimes of low differentiability in general relativity.