This article deals with vector valued differential forms on $C^\infty$-manifolds. As a generalization of the exterior product, we introduce an operator that combines $\operatorname{Hom}(\bigotimes^s(W),Z)$-valued forms with $\operatorname{Hom}(\bigotimes^s(V),W)$-valued forms. We discuss the main properties of this operator such as (multi)linearity, associativity and its behavior under pullbacks, push-outs, exterior differentiation of forms, etc. Finally we present applications for Lie groups and fiber bundles.
@article{118957,
author = {Christian Gross},
title = {A generalization of the exterior product of differential forms combining Hom-valued forms},
journal = {Commentationes Mathematicae Universitatis Carolinae},
volume = {38},
year = {1997},
pages = {587-602},
zbl = {0946.58002},
mrnumber = {1485080},
language = {en},
url = {http://dml.mathdoc.fr/item/118957}
}
Gross, Christian. A generalization of the exterior product of differential forms combining Hom-valued forms. Commentationes Mathematicae Universitatis Carolinae, Tome 38 (1997) pp. 587-602. http://gdmltest.u-ga.fr/item/118957/
Connections, Curvature, and Cohomology, Vol. II, Academic Press, 1973. | Zbl 0372.57001
Operators on differential forms for Lie transformation groups, Journal of Lie Theory 6 (1996), 1-17. (1996) | MR 1406002 | Zbl 0872.58001
Connections on fiber bundles and canonical extensions of differential forms, http://www.mathematik.th-darmstadt.de/prepr, Preprint No. 1795.
Cohomology and connections on fiber bundles and applications to field theories, Journal of Mathematical Physics 37 12 (1996), 6375-6394. (1996) | MR 1419176 | Zbl 0863.53017
Differential Geometry and Symmetric Spaces, Academic Press, 1962. | MR 0145455 | Zbl 0122.39901
Foundations of Differential Geometry, Vol. I, John Wiley & Sons, 1963. | MR 1393940 | Zbl 0526.53001