Forms Equivalent to Curvatures

Abstract

The 2-forms, $\Omega$ and ${\Omega }^{\prime }$ on a manifold $M$ with values in vector bundles $\xi \to M$ and ${\xi }^{\prime }\to M$ are $equivalent$ if there exist smooth fibered-linear maps $U:\xi \to {\xi }^{\prime }$ and $W:{\xi }^{\prime }\to \xi$ with ${\Omega }^{\prime }=U\Omega$ and $\Omega =W{\Omega }^{\prime }$. 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.