Vector-valued differential form: Difference between revisions

Revision as of 00:58, 15 December 2014

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Given ''P'' and ''ρ'' as above one can construct the [[associated vector bundle]] ''E'' = ''P'' &times;_''ρ'' ''V''. Tensorial ''q''-forms on ''P'' are in a natural one-to-one correspondence with ''E''-valued ''q''-forms on ''M''. As in the case of the principal bundle F(''E'') above, ''E''-valued forms on ''M'' pull back to ''V''-valued forms on ''P''. Explicitly, given a ''q''-form <math>\overline{\phi}</math>, define φ fiberwise by (say at ''u'')

:<math>\phi = u^{-1}\pi^*\overline{\phi}</math>

where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_xE_{\pi(u)} = (\pi^*E)_u</math>. φ is then a tensorial form of type ρ. Conversely, given a tensorial form φ of type ρ, the same formula defines an ''E''-valued form <math>\overline{\phi}</math> on ''M''. In particular, there is a natural bijection

:<math>\Gamma(M, E) \to \{ f: P \to V | f(ug) = g^{-1}f(u) \}</math>.