Nonlinear generalized functions on manifolds

Abstract

In this work, we adopt a new approach to the construction of a global theory of algebras of generalized functions on manifolds based on the concept of smoothing operators. This produces a generalization of previous theories in a form which is suitable for applications to differential geometry. The generalized Lie derivative is introduced and shown to extend the Lie derivative of Schwartz distributions. A new feature of this theory is the ability to define a covariant derivative of generalized scalar fields which extends the covariant derivative of distributions at the level of association. We end by sketching some applications of the theory. This work also lays the foundations for a nonlinear theory of distributional geometry that is developed in a subsequent paper that is based on Colombeau algebras of tensor distributions on manifolds.

A nonlinear theory of distributional geometry

“…However, despite recent progress [16] these methods are not yet well developed for Lorentzian metrics and applications to general relativity. In this paper we will adopt a different approach using the theory of nonlinear generalized functions as described in our previous paper [17].…”

“…We will continue to use the notation of [17]. In particular, Xfalse(Mfalse) and Ω p ( M ) denote the spaces of smooth vector fields and p -forms on M , respectively.…”

A nonlinear theory of distributional geometry

“…In a previous paper [1] we introduced a global theory of generalised functions on a manifold M. The key idea was to replace a nonsmooth function f by 1-parameter families of smooth functions according to (1) fε (x) = M f (y)ω x,ε (y), depending on a suitable family of smoothing kernels (ω ε ) ε . For fixed ε these may be treated just like smooth functions on manifolds so all the standard operations that may be carried out on smooth functions extend to the smoothed functions fε .…”

“…For fixed ε these may be treated just like smooth functions on manifolds so all the standard operations that may be carried out on smooth functions extend to the smoothed functions fε . The embedding (1) extends to distributions T ∈ D ′ (M) by defining (2) T (ω ε )(x) = T, ω x,ε .…”

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This paper builds on the theory of generalised functions begun in [1]. The Colombeau theory of generalised scalar fields on manifolds is extended to a nonlinear theory of generalised tensor fields which is diffeomorphism invariant and has the sheaf property.

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