Vector-valued differential form: Difference between revisions

Revision as of 21:59, 15 December 2014

Revision as of 22:01, 15 December 2014

Line 66:

==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'') ×_\rho '''R'''^''k'' via the inverse of [''u'', ''v''] →''u''(''v''), where ρ is the standard representation. 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