Geometric properties of involutive distributions on graded manifolds
The authors are working here in the category of \(\mathbb Z _2\)-graded manifolds (equivalently supermanifolds, in the sense of Berezin-Leites-Kostant). As one of their main results, they prove an adaptation of the Frobenius theorem to \(\mathbb Z _2\)-graded submersions. Among other results, they also show that, roughly speaking, an involutive distribution \(\mathcal D\) on a \(\mathbb Z _2\)-graded manifold \((M, \mathcal A)\) gives rise to a foliation \(\mathcal F\) of \(M\), and that the space of leaves \(M/\mathcal F\) possesses a \(\mathbb Z _2\)-graded manifold structure if and only if \(\mathcal F\) is regular. The \(\mathbb Z _2\)-graded integral submanifolds of \(\mathcal D\) are studied. The results on involutive distributions are then used to characterize split \(\mathbb Z _2\)-graded manifolds over an orientable base. Split \(\mathbb Z _2\)-graded manifolds with Batchelor's bundle having a trivial direct summand are also characterized. As applications, the authors present several results on graded Lie groups and graded Lie algebras.
scientific article; zbMATH DE number 1052951
scientific article