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m →Basic or tensorial forms on principal bundles: Fixing links to disambiguation pages, improving links, other minor cleanup tasks |
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Let ''M'' be a [[smooth manifold]] and ''E'' → ''M'' be a smooth [[vector bundle]] over ''M''. We denote the space of [[section (fiber bundle)|smooth section]]s of a bundle ''E'' by Γ(''E''). An '''''E''-valued differential form''' of degree ''p'' is a smooth section of the [[tensor product bundle]] of ''E'' with Λ<sup>''p''</sup>(''T''<sup> ∗</sup>''M''), the ''p''-th [[exterior power]] of the [[cotangent bundle]] of ''M''. The space of such forms is denoted by
:<math>\Omega^p(M,E) = \Gamma(E\otimes\Lambda^pT^*M).</math>
Because Γ is a [[strong monoidal functor]],<ref name=gamma_monoidal>{{cite web|title=Global sections of a tensor product of vector bundles on a smooth manifold|url=
:<math>\Gamma(E\otimes\Lambda^pT^*M) = \Gamma(E) \otimes_{\Omega^0(M)} \Gamma(\Lambda^pT^*M) = \Gamma(E) \otimes_{\Omega^0(M)} \Omega^p(M),</math>
where the latter two tensor products are the [[tensor product of modules]] over the [[ring (mathematics)|ring]] Ω<sup>0</sup>(''M'') of smooth '''R'''-valued functions on ''M'' (see the seventh example [[module (mathematics)#Examples|here]]). By convention, an ''E''-valued 0-form is just a section of the bundle ''E''. That is,
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