Previous |  Up |  Next


differential forms; exterior product; multilinear algebra
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.
[1] Greub W., Halperin S., Vanstone R.: Connections, Curvature, and Cohomology. Vol. II, Academic Press, 1973. Zbl 0372.57001
[2] Gross C.: Operators on differential forms for Lie transformation groups. Journal of Lie Theory 6 (1996), 1-17. MR 1406002 | Zbl 0872.58001
[3] Gross C.: Connections on fiber bundles and canonical extensions of differential forms., Preprint No. 1795.
[4] Gross C.: Cohomology and connections on fiber bundles and applications to field theories. Journal of Mathematical Physics 37 12 (1996), 6375-6394. MR 1419176 | Zbl 0863.53017
[5] Helgason S.: Differential Geometry and Symmetric Spaces. Academic Press, 1962. MR 0145455 | Zbl 0122.39901
[6] Kobayashi S., Nomizu K.: Foundations of Differential Geometry. Vol. I, John Wiley & Sons, 1963. MR 1393940 | Zbl 0526.53001
Partner of
EuDML logo