Content deleted Content added
TakuyaMurata (talk | contribs) |
TakuyaMurata (talk | contribs) |
||
Line 79:
where ''u'' is viewed as a linear isomorphism <math>V \overset{\simeq}\to E_{\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'' (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>.
Example: Let ''E'' be the tangent bundle of ''M''. Then identity bundle map id<sub>''E''</sub>: ''E'' →''E'' is an ''E''-valued one form on ''M''. The [[canonical one-form]] is a unique one-form on the frame bundle of ''E'' that corresponds to id<sub>''E''</sub>.
==Notes==
| |||