Vector-valued differential form: Difference between revisions

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==Basic or tensorial forms on principal bundles==

Let ''E'' → ''M'' be a smooth vector bundle of rank ''k'' over ''M'' and let ''π'' : F(''E'') → ''M'' be the ([[associated bundle|associated]]) [[frame bundle]] of ''E'', which is a [[principal bundle|principal]] GL_''k''('''R''') bundle over ''M''. The [[pullback bundle|pullback]] of ''E'' by ''π'' is canonically isomorphic to the trivial bundle F(''E'') × '''R'''^''k'' via (the inverse of) [''u'', ''v''] →''u''(''v''). Therefore, the pullback by ''π'' of an ''E''-valued form on ''M'' determines an '''R'''^''k''-valued form on F(''E''). It is not hard to check that this pulled back form is [[equivariant|right-equivariant]] with respect to the natural [[group action|action]] of GL_''k''('''R''') on F(''E'') × '''R'''^''k'' and vanishes on [[vertical bundle|vertical vectors]] (tangent vectors to F(''E'') which lie in the kernel of d''π''). Such vector-valued forms on F(''E'') are important enough to warrant special terminology: they are called ''basic'' or ''tensorial forms'' on F(''E'').

Let ''π'' : ''P'' → ''M'' be a (smooth) [[principal bundle|principal ''G''-bundle]] and let ''V'' be a fixed vector space together with a [[group representation|representation]] ''ρ'' : ''G'' → GL(''V''). A '''basic''' or '''tensorial form''' on ''P'' of type ρ is a ''V''-valued form ω on ''P'' which is '''equivariant''' and '''horizontal''' in the sense that

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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, given a ''q''-form <math>\overline{\phi}</math> on ''M'', define φ on ''P'' 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_{\pi(u)} = (\pi^*E)_u, v \mapsto [u, v]</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'' (cf. the [[Chern–Weil homomorphism]].) In particular, there is a natural isomorphism of vector spaces

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