Vector-valued differential form: Difference between revisions

Revision as of 21:59, 17 December 2014

Revision as of 22:01, 17 December 2014

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:<math>\wedge : \Omega^p(M,E_1) \times \Omega^q(M,E_2) \to \Omega^{p+q}(M,E_1\otimes E_2).</math>

The definition is just as for ordinary forms with the exception that real multiplication is replaced with the [[tensor product]]:

:<math>(\omega\wedge\eta)(v_1,\cdots,v_{p+q}) = \frac{1}{(p + q)!}\sum_{\pisigma\in S_{p+q}}\sgn(\pisigma)\omega(v_{\pisigma(1)},\cdots,v_{\pisigma(p)})\otimes \eta(v_{\pisigma(p+1)},\cdots,v_{\pisigma(p+q)}).</math>

In particular, the wedge product of an ordinary ('''R'''-valued) ''p''-form with an ''E''-valued ''q''-form is naturally an ''E''-valued (''p''+''q'')-form (since the tensor product of ''E'' with the trivial bundle ''M'' &times; '''R''' is [[naturally isomorphic]] to ''E''). For ω ∈ Ω^''p''(''M'') and η ∈ Ω^''q''(''M'', ''E'') one has the usual commutativity relation:

:<math>\omega\wedge\eta = (-1)^{pq}\eta\wedge\omega.</math>