# Article

 Title: A generalization of the exterior product of differential forms combining Hom-valued forms (English) Author: Gross, Christian Language: English Journal: Commentationes Mathematicae Universitatis Carolinae ISSN: 0010-2628 (print) ISSN: 1213-7243 (online) Volume: 38 Issue: 3 Year: 1997 Pages: 587-602 . Category: math . Summary: 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. (English) Keyword: differential forms Keyword: exterior product Keyword: multilinear algebra MSC: 15A75 MSC: 22E30 MSC: 53C05 MSC: 58A05 MSC: 58A10 MSC: 58A15 idZBL: Zbl 0946.58002 idMR: MR1485080 . Date available: 2009-01-08T18:36:41Z Last updated: 2012-04-30 Stable URL: http://hdl.handle.net/10338.dmlcz/118957 . Reference: [1] Greub W., Halperin S., Vanstone R.: Connections, Curvature, and Cohomology.Vol. II, Academic Press, 1973. Zbl 0372.57001 Reference: [2] Gross C.: Operators on differential forms for Lie transformation groups.Journal of Lie Theory 6 (1996), 1-17. Zbl 0872.58001, MR 1406002 Reference: [3] Gross C.: Connections on fiber bundles and canonical extensions of differential forms.http://www.mathematik.th-darmstadt.de/prepr, Preprint No. 1795. Reference: [4] Gross C.: Cohomology and connections on fiber bundles and applications to field theories.Journal of Mathematical Physics 37 12 (1996), 6375-6394. Zbl 0863.53017, MR 1419176 Reference: [5] Helgason S.: Differential Geometry and Symmetric Spaces.Academic Press, 1962. Zbl 0122.39901, MR 0145455 Reference: [6] Kobayashi S., Nomizu K.: Foundations of Differential Geometry.Vol. I, John Wiley & Sons, 1963. Zbl 0526.53001, MR 1393940 .

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