Pullback of Differential Forms
A particular important case of the pullback of covariant tensor fields is the pullback of differential forms. If α is a differential k-form, i.e., a section of the exterior bundle ΛkT*N of (fiberwise) alternating k-forms on TN, then the pullback of α is the differential k-form on M defined by the same formula as in the previous section:
for x in M and Xj in TxM.
The pullback of differential forms has two properties which make it extremely useful.
1. It is compatible with the wedge product in the sense that for differential forms α and β on N,
2. It is compatible with the exterior derivative d: if α is a differential form on N then
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