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A nonlinear theory of distributional geometry

A nonlinear theory of distributional geometry
A nonlinear theory of distributional geometry
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 Cmetrics 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, tensor fields, distributional covariant derivative, distributional geometry, Colombeau algebra, diffeomorphism-invariant
1364-5021
1-20
Nigsch, Eduard A.
ea9b17e3-8ab7-4222-b1b3-a9e0e428634e
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634
Nigsch, Eduard A.
ea9b17e3-8ab7-4222-b1b3-a9e0e428634e
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634

Nigsch, Eduard A. and Vickers, James A. (2020) A nonlinear theory of distributional geometry. Proceedings of the Royal Society A, 476, 1-20, [2020.0642]. (doi:10.1098/rspa.2020.0642).

Record type: Article

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 Cmetrics 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.

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1910.03426 - Author's Original
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NonlinearGenFunctions - Accepted Manuscript
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More information

In preparation date: 8 October 2019
Accepted/In Press date: 16 November 2020
e-pub ahead of print date: 16 December 2020
Published date: 23 December 2020
Additional Information: This is the published version which replaces the ArXiv version of the same name. The file DistributionalGeom.pdf is the "accepted manuscript"
Keywords: nonlinear generalized functions, tensor fields, distributional covariant derivative, distributional geometry, Colombeau algebra, diffeomorphism-invariant

Identifiers

Local EPrints ID: 435183
URI: http://eprints.soton.ac.uk/id/eprint/435183
ISSN: 1364-5021
PURE UUID: d23a18ef-c0df-4638-a30f-8d346377a2b5
ORCID for James A. Vickers: ORCID iD orcid.org/0000-0002-1531-6273

Catalogue record

Date deposited: 25 Oct 2019 16:30
Last modified: 18 Feb 2021 16:33

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