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Forms equivalent to curvatures. (English) Zbl 0632.53031

Two 2-forms \(\Omega\) and \(\Omega\) ’ on a manifold M with values in vector bundles \(\xi\to M\) and \(\xi\) ’\(\to M\) are said to be equivalent, if there exist smooth fibered-linear maps \(U: \xi\to \xi '\) and \(W: \xi\) ’\(\to \xi\) such that \(\Omega '=U\Omega\) and \(\Omega =W\Omega '\). Near a point \(x\in M\), a 2-form \(\omega\) is said to have an integrating factor f, if \(f\omega\) is closed near x. Let \(\theta\to M\) be a bundle endowed with a connection. Now a theorem, for example, states that: a 2-form \(\omega\) equivalent to the curvature has an integrating factor near x, if \(rank(\omega)=2\) near x or \(rank(\omega)>2\) near x. Some other theorems and propositions are stated, which are also related to the local or global integrating factors.
Reviewer: H.Wakakuwa

MSC:

53C05 Connections (general theory)
58A10 Differential forms in global analysis
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