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TakuyaMurata (talk | contribs) →Basic or tensorial forms on principal bundles: connection |
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:<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 [[tautological one-form]] is a unique one-form on the frame bundle of ''E'' that corresponds to id<sub>''E''</sub>. Denoted by θ, it is a tensorial form of standard type.<!--Mention this somewhere else: The [[exterior covariant derivative]] of θ, Θ = ''D''θ is called a [[torsion form]].-->
Now, suppose there is a connection on ''P'' so that there is an [[exterior covariant differentiation]] ''D'' on (various) vector-valued forms on ''P''. Through the above correspondence, ''D'' also acts on ''E''-valued forms: define ∇ by
:<math>\nabla \overline{\phi} = \overline{D \phi}</math>
In particular for zero-forms,
:<math>\nabla: \Gamma(M, E) \to \Gamma(M, T^*M \otimes E)</math>.
This is exactly the [[covariant derivative]] for the [[connection (vector bundle)|connection on the vector bundle ''E'']]. (Proof: <math>D (f\phi) = Df \otimes \phi + f D\phi</math> for any scalar-valued tensorial zero-form ''f'' and any tensorial zero-form φ of type ρ, and ''Df'' = ''df'' since ''f'' descends to a function on ''M''; cf. this [[Chern–Weil homomorphism#Definition of the homomorphism|Lemma 2]].)
==Notes==
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