A synthetic Frobenius theorem (Q1295521)

The author presents some aspects concerning the extension of the classical Frobenius theorem from the theory of completely integrable exterior differential systems, in the context of synthetic differential geometry. The objects in the synthetic differential geometry are the smooth spaces including some more general spaces than the \(C^\infty\) manifolds. This extension allows to form function spaces, quotient spaces, the spaces of zeroes of arbitary smooth functions, etc. A differential \(1\)-form on a smooth space \(M\) may be thought of as a map from \(M\) into a suitable universal space. Let \(\omega ^1, \dots , \omega ^n\) be a set of \(1\)-forms on the smooth space \(M\), satisfying the well-known involutivity condition from the classical Frobenius theorem. Denote by \({\mathcal F}\) the distribution in \(TM\) defined by \(\omega ^1,\dots , \omega ^n\). The forms \(\omega ^1,\dots , \omega ^n\) may be thought of as functions \(\hat\omega ^1,\dots ,\hat\omega ^n\) taking values in the universal space. Let \(F\subset M\) be the set of their common zeroes. Theorem 1.1. \(TF= F\times_M {\mathcal F}\). Theorem 1.2. If \(M\) is Frobenius microlinear, then \(F\) is Frobenius microlinear. Section 2 of the paper contains a rather detailed introduction to synthetic differential geometry.