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Let \(\pi :E\rightarrow M\) be a smooth \({\mathbb K}\) -vector bundle. Denote by \(A^p(E)\) the space of sections of the bundle \(\Lambda ^{p}T^{*}M\otimes E\) ( \(\text {where}\;A^0(E)\) is the space of the sections of E), i.e., \(A^p(E)\) is the space of p-forms on M with values in the bundle E. The ring of smooth functions on M will be denoted by \(\mathcal {F}(M)\) , and the space of vector fields on M by \(\chi (M)\) ; \(\chi (M)\) is a module over \(\mathcal {F}(M)\) .

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Connections on Vector Bundles

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  • 发表日期：

    2026年

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  • Johann Davidov

摘要

Let \(\pi :E\rightarrow M\) be a smooth \({\mathbb K}\) -vector bundle. Denote by \(A^p(E)\) the space of sections of the bundle \(\Lambda ^{p}T^{*}M\otimes E\) ( \(\text {where}\;A^0(E)\) is the space of the sections of E), i.e., \(A^p(E)\) is the space of p-forms on M with values in the bundle E. The ring of smooth functions on M will be denoted by \(\mathcal {F}(M)\) , and the space of vector fields on M by \(\chi (M)\) ; \(\chi (M)\) is a module over \(\mathcal {F}(M)\) .