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Sheaves of nonlinear generalized functions and manifold-valued distributions

Sheaves of nonlinear generalized functions and manifold-valued distributions
Sheaves of nonlinear generalized functions and manifold-valued distributions
This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space G[X,Y] of Colombeau generalized functions defined on a manifold X and taking values in a manifold Y. This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings C(X,Y) into G[X,Y] and study the sheaf properties of G[X,Y]. Similar results are obtained for spaces of generalized vector bundle homomorphisms. Based on these constructions we propose the definition of a space D'[X,Y] of distributions on X taking values in Y. D'[X,Y] is realized as a quotient of a certain subspace of G[X,Y]
algebras of generalized functions, colombeau algebras, generalized functions on manifolds, manifold-valued distributions
0002-9947
5177-5192
Kunzinger, Michael
5ee9f681-a923-4fb5-b1c8-9454237dd721
Steinbauer, Roland
053836b9-b9d0-4a1d-93b1-06500bc87b17
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634
Kunzinger, Michael
5ee9f681-a923-4fb5-b1c8-9454237dd721
Steinbauer, Roland
053836b9-b9d0-4a1d-93b1-06500bc87b17
Vickers, James A.
719cd73f-c462-417d-a341-0b042db88634

Kunzinger, Michael, Steinbauer, Roland and Vickers, James A. (2009) Sheaves of nonlinear generalized functions and manifold-valued distributions. Transactions of the American Mathematical Society, 361, 5177-5192. (doi:10.1090/S0002-9947-09-04621-2).

Record type: Article

Abstract

This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space G[X,Y] of Colombeau generalized functions defined on a manifold X and taking values in a manifold Y. This space is essential in order to study concepts such as flows of generalized vector fields or geodesics of generalized metrics. We introduce an embedding of the space of continuous mappings C(X,Y) into G[X,Y] and study the sheaf properties of G[X,Y]. Similar results are obtained for spaces of generalized vector bundle homomorphisms. Based on these constructions we propose the definition of a space D'[X,Y] of distributions on X taking values in Y. D'[X,Y] is realized as a quotient of a certain subspace of G[X,Y]

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Published date: 2009
Keywords: algebras of generalized functions, colombeau algebras, generalized functions on manifolds, manifold-valued distributions
Organisations: Applied Mathematics

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Local EPrints ID: 66525
URI: http://eprints.soton.ac.uk/id/eprint/66525
ISSN: 0002-9947
PURE UUID: 4cf21dec-cdcb-4261-91c3-6e5937567d45
ORCID for James A. Vickers: ORCID iD orcid.org/0000-0002-1531-6273

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Date deposited: 29 Jun 2009
Last modified: 09 Jan 2022 02:33

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Contributors

Author: Michael Kunzinger
Author: Roland Steinbauer

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