Vector-valued differential form: Difference between revisions

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Let ''V'' be a fixed [[vector space]]. A '''''V''-valued differential form''' of degree ''p'' is a differential form of degree ''p'' with values in the [[trivial bundle]] ''M'' × ''V''. The space of such forms is denoted Ω^''p''(''M'', ''V''). When ''V'' = '''R''' one recovers the definition of an ordinary differential form. If ''V'' is finite-dimensional, then one can show that the natural homomorphism

:<math>\Omega^p(M) \otimes_\mathbb{R} V \to \Omega^p(M,V),</math>

where the first tensor product is of vector spaces over '''R''', is an isomorphism. <ref>Proof: One can verify this for ''p''=0 by turning a basis for ''V'' into a set of constant functions to ''V'', which allows the construction of an inverse to the above homomorphism. The general case can be proved by noting that

:<math>\Omega^p(M, V) = \Omega^0(M, V) \otimes_{\Omega^0(M)} \Omega^p(M),</math>

and that because <math>\mathbb{R}</math> is a sub-ring of Ω⁰(''M'') via the constant functions,

:<math>\Omega^0(M, V) \otimes_{\Omega^0(M)} \Omega^p(M) = (V \otimes_\mathbb{R} \Omega^0(M)) \otimes_{\Omega^0(M)} \Omega^p(M) = V \otimes_\mathbb{R} (\Omega^0(M) \otimes_{\Omega^0(M)} \Omega^p(M)) = V \otimes_\mathbb{R} \Omega^p(M).</math></ref>

==Operations on vector-valued forms==