Vector-valued differential form: Difference between revisions

Revision as of 00:00, 9 August 2019

Revision as of 10:35, 26 August 2019

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:<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'']].<ref>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]].</ref>

==Examples==

[[Siegel modular form]]s arise as vector-valued differential forms on [[Siegel modular variety|Siegel modular varieties]].<ref>{{cite journal|title=The Geometry of Siegel Modular Varieties |last1=Hulek |first1=Klaus |last2=Sankaran |first2=G. K. |journal=Advanced Studies in Pure Mathematics |volume=35 |year=2002 |pages=89–156}}</ref>

==Notes==