Revista Matemática Iberoamericana

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Volume 2, Issue 4, 1986, pp. 397–403
DOI: 10.4171/RMI/41

Published online: 1986-12-31

Forms Equivalent to Curvatures

Horacio Porta[1] and Lázaro Recht[2]

(1) University of Illinois, Urbana, USA
(2) Universidad Simón Bolívar, Caracas, Venezuela

The 2-forms, $\Omega$ and $\Omega '$ on a manifold $M$ with values in vector bundles $\xi \rightarrow M$ and $\xi ' \rightarrow M$ are $equivalent$ if there exist smooth fibered-linear maps $U: \xi \rightarrow \xi '$ and $W: \xi ' \rightarrow \xi$ with $\Omega ' = U\Omega$ and $\Omega = W\Omega '$. It is shown that an ordinary 2-form equivalent to the curvature of a linear connection has locally a non-vanishing integrating factor at each point in the interior of the set rank $(\omega) = 2$ or in the set rank $(\omega) > 2$. Under favorable conditions the same holds at points where the rank of $\omega$ changes from =2 to >2. Global versions are also considered.

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Porta Horacio, Recht Lázaro: Forms Equivalent to Curvatures. Rev. Mat. Iberoam. 2 (1986), 397-403. doi: 10.4171/RMI/41