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Sheaves of nonlinear generalized functions and manifold-valued distributions


Authors: Michael Kunzinger, Roland Steinbauer and James A. Vickers
Journal: Trans. Amer. Math. Soc. 361 (2009), 5177-5192
MSC (2000): Primary 46T30; Secondary 46F30, 53B20
DOI: https://doi.org/10.1090/S0002-9947-09-04621-2
Published electronically: April 21, 2009
MathSciNet review: 2515808
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Abstract: This paper is part of an ongoing program to develop a theory of generalized differential geometry. We consider the space $ \mathcal{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 $ \mathcal{C}(X,Y)$ into $ \mathcal{G}[X,Y]$ and study the sheaf properties of $ \mathcal{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 $ \mathcal{D}'[X,Y]$ of distributions on $ X$ taking values in $ Y$. $ \mathcal{D}'[X,Y]$ is realized as a quotient of a certain subspace of $ \mathcal{G}[X,Y]$.


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Additional Information

Michael Kunzinger
Affiliation: Department of Mathematics, University of Vienna, Nordbergstr. 15, A-1090 Wien, Austria
Email: michael.kunzinger@univie.ac.at

Roland Steinbauer
Affiliation: Department of Mathematics, University of Vienna, Nordbergstr. 15, A-1090 Wien, Austria
Email: roland.steinbauer@univie.ac.at

James A. Vickers
Affiliation: Faculty of Mathematical Studies, University of Southampton, Highfield, Southampton, SO17 1BJ, United Kingdom
Email: J.A.Vickers@maths.soton.ac.uk

DOI: https://doi.org/10.1090/S0002-9947-09-04621-2
Keywords: Algebras of generalized functions, Colombeau algebras, generalized functions on manifolds, manifold-valued distributions.
Received by editor(s): April 16, 2007
Received by editor(s) in revised form: August 24, 2007
Published electronically: April 21, 2009
Additional Notes: This work was supported by project P16742 and START-project Y-237 of the Austrian Science Fund
Article copyright: © Copyright 2009 American Mathematical Society

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