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Nonlinear generalized functions on manifolds

Nonlinear generalized functions on manifolds
Nonlinear generalized functions on manifolds
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.
nonlinear generalized functions, Colombeau algebra, diffeomorphism invariant
1364-5021
20200640
Vickers, James
719cd73f-c462-417d-a341-0b042db88634
Nigsch, Eduard A.
ea9b17e3-8ab7-4222-b1b3-a9e0e428634e
Vickers, James
719cd73f-c462-417d-a341-0b042db88634
Nigsch, Eduard A.
ea9b17e3-8ab7-4222-b1b3-a9e0e428634e

Vickers, James and Nigsch, Eduard A. (2020) Nonlinear generalized functions on manifolds. Proceedings of the Royal Society A, 476 (2244), 20200640, [20200640]. (doi:10.1098/rspa.2020.0640).

Record type: Article

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.

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1910.03411 - Accepted Manuscript
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Accepted/In Press date: 16 November 2020
e-pub ahead of print date: 16 December 2020
Published date: December 2020
Additional Information: This is the published version and replaces the ArXiv version of the paper with the same title. The file NonlinearGenFunctions.pdf is the "accepted manuscript"
Keywords: nonlinear generalized functions, Colombeau algebra, diffeomorphism invariant

Identifiers

Local EPrints ID: 446564
URI: http://eprints.soton.ac.uk/id/eprint/446564
ISSN: 1364-5021
PURE UUID: 0784aab5-905c-4ef1-99fb-0e39af1e1209
ORCID for James Vickers: ORCID iD orcid.org/0000-0002-1531-6273

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Date deposited: 15 Feb 2021 17:31
Last modified: 17 Mar 2024 02:32

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Contributors

Author: James Vickers ORCID iD
Author: Eduard A. Nigsch

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