A nonlinear theory of distributional geometry

Abstract

This paper builds on the theory of nonlinear generalized functions begun in Nigsch & Vickers (Nigsch, Vickers 2021 Proc. R. Soc. A 20200640 ( doi:10.1098/rspa.2020.0640 )) and extends this to a diffeomorphism-invariant nonlinear theory of generalized tensor fields with the sheaf property. The generalized Lie derivative is introduced and shown to commute with the embedding of distributional tensor fields and the generalized covariant derivative commutes with the embedding at the level of association. The concept of a generalized metric is introduced and used to develop a non-smooth theory of differential geometry. It is shown that the embedding of a continuous metric results in a generalized metric with well-defined connection and curvature and that for C 2 metrics the embedding preserves the curvature at the level of association. Finally, we consider an example of a conical metric outside the Geroch–Traschen class and show that the curvature is associated to a delta function.

Nonlinear generalized functions on manifolds

“…The latter is simply the ordinary Lie derivative of a smooth function and therefore agrees with the directional derivative or covariant derivative of a function. This will allow us to define the covariant derivative of a generalized tensor field in [19]. Although the ordinary Lie derivative does not commute with the embedding of distributions, as is the case on Rn, it does so at the level of association.…”

“…In this context, the flow given by the generalized vector field F is a solution of the system of ODEs ddtnormalΦfalse(t,xfalse)=Ffalse(normalΦfalse(t,xfalse)false)1eminscriptGfalse[R1+n,Rnfalse] and normalΦfalse(0,⋅false)=idRn1eminscriptGfalse[Rn,Rnfalse]. However, to obtain generalized solutions to (classical) singular ODEs requires a canonical embedding into the algebra as provided by the present theory. Furthermore, considering flows and generalized vector fields F on manifolds M will be possible using the theory developed in [19].…”

Nonlinear generalised functions on manifolds

“…In section 3 we will present a new version of the algebra based on the idea of smoothing operators. This has a larger basic space than [12] which allows us to define a covariant derivative and can therefore be developed into a nonlinear theory of distributional differential geometry [1]. In contrast to the theory on R n the theory of generalised functions on manifolds involves a number of technical issues involving in particular the theory of differentiation in locally convex spaces.…”

“…Because the embedding into the algebra does not commute with multiplication (except on the subalgebra of smooth functions) one cannot simply work with the coordinate components of a tensor and use the theory of generalised scalars. In a subsequent paper [1] we show how it is possible to define an algebra of generalised tensor fields on a manifold which contains the spaces of smooth tensor fields as a subalgebra and has a canonical coordinate independent embedding of the spaces of tensor distributions as linear subspaces.…”

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This paper lays the foundations for a nonlinear theory of differential geometry that is developed in a subsequent paper [1] 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.

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