Sheaves of nonlinear generalized functions and manifold-valued distributions
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- by Michael Kunzinger, Roland Steinbauer and James A. Vickers PDF
- Trans. Amer. Math. Soc. 361 (2009), 5177-5192 Request permission
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 $\mathcal {G}[X,Y]$.References
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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
- 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
- © Copyright 2009 American Mathematical Society
- 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
- MathSciNet review: 2515808