Vector-valued differential form: Difference between revisions

Revision as of 22:50, 21 September 2021

Latest revision as of 07:37, 12 April 2025

(2 intermediate revisions by 2 users not shown)

Line 37:

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_{\sigma\in S_{p+q}}\sgn(\sigma)\omega(v_{\sigma(1)},\cdots,v_{\sigma(p)})\otimes \eta(v_{\sigma(p+1)},\cdots,v_{\sigma(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:

In terms of local frames {''e''_''α''} and {''l''_''β''} for ''E''₁ and ''E''₂ respectively, the wedge product of an ''E''₁-valued ''p''-form ''ω'' = ''ω''^''α'' ''e''_''α'', and an ''E''₂-valued ''q''-form ''η'' = ''η''^''β'' ''l''_''β'' is

:<math>\omega \wedge \eta = \sum_{\alpha, \beta} (\omega^\alpha \wedge \eta^\beta) (e_\alpha \otimes l_\beta),</math>

where ''ω''^''α'' ∧ ''η''^''β'' is the ordinary wedge product of <math>\mathbb{R}</math>-valued forms.

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>

Line 68 ⟶ 72:

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 F(''E'') &times;_ρ '''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 (mathematics)|action]] of GL_''k''('''R''') on F(''E'') &times; '''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'' whichthat is '''equivariant''' and '''horizontal''' in the sense that

#<math>(R_g)^*\omega = \rho(g^{-1})\omega\,</math> for all ''g'' ∈ ''G'', and

#<math>\omega(v_1, \ldots, v_p) = 0</math> whenever at least one of the ''v''_''i'' are vertical (i.e., d''π''(''v''_''i'') = 0).