Tensor Valued Colombeau Functions on Manifolds

TL;DR

This paper develops a framework for tensor-valued Colombeau functions on manifolds, extending scalar cases and enabling tensor distributions and Lie derivatives within a generalized function algebra.

Contribution

It introduces a new basic space for tensor-valued Colombeau functions on manifolds, incorporating transport operators and extending classical differential geometric tools.

Findings

Constructed a basic space containing tensor distributions

Enabled natural extension of Lie derivatives for tensor-valued functions

Provided a foundation for tensor-valued Colombeau algebra on manifolds

Abstract

Extending the construction of the (intrinsically defined) full algebra of scalar valued Colombeau functions on a smooth manifold M (Grosser et al., Adv. Math. 166 (2002), 179-206) we present a suitable basic space for eventually obtaining tensor valued generalized functions on M, via the usual quotient construction. This basic space canonically contains the tensor valued distributions and permits a natural extension of the classical Lie derivative. Its members are smooth functions depending - via a third slot - on so-called transport operators, in addition to slots one (smooth n-forms on M) and two (points of M) from the scalar case.