Nonlinear tensor distributions on Riemannian manifolds

TL;DR

This paper develops a nonlinear algebra of tensor distributions on Riemannian manifolds, enabling the embedding of distributional tensors into a Colombeau-type algebra while maintaining key geometric properties.

Contribution

It introduces a simplified construction of a Colombeau algebra of tensor fields on manifolds using a background connection, extending scalar theories to tensor distributions.

Findings

Embedding of tensor distributions into the algebra is canonical.

The construction simplifies previous approaches by leveraging existing scalar theories.

The embedding commutes with certain geometric operations on Riemannian manifolds.

Abstract

We construct an algebra of nonlinear generalized tensor fields on manifolds in the sense of J.-F. Colombeau, i.e., containing distributional tensor fields as a linear subspace and smooth tensor fields as a faithful subalgebra. The use of a background connection on the manifold allows for a simplified construction based on the existing scalar theory of full diffeomorphism invariant Colombeau algebras on manifolds, still having a canonical embedding of tensor distributions. In the particular case of the Levi-Civita connection on Riemannian manifolds one obtains that this embedding commutes with pullback along homotheties and Lie derivatives along Killing vector fields only.