Geometric properties of involutive distributions on graded manifolds
type: Item
instance of: scholarly article
title: Geometric properties of involutive distributions on graded manifolds
zbMATH Open document ID: 0887.58008
DOI: 10.1016/S0019-3577(97)89122-7
published in: Indagationes Mathematicae. New Series
publication date: 1998-05-06T00:00:00Z
review text: 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.
Mathematics Subject Classification ID: 58C50
Mathematics Subject Classification ID: 58A50
Mathematics Subject Classification ID: 57R30
zbMATH DE Number: 1052951
zbMATH Keywords: graded manifold
zbMATH Keywords: supermanifold
zbMATH Keywords: involutive distribution
zbMATH Keywords: graded submersion
zbMATH Keywords: Frobenius theorem
zbMATH Keywords: foliation
zbMATH Keywords: Batchelor bundle
zbMATH Keywords: graded ordinary differential equation
zbMATH Keywords: graded Lie group
zbMATH Keywords: graded Lie algebra
zbMATH Keywords: graded vector field
author: Juan Monterde
author: Jaime Muñoz Masqué
author: O. A. Sánchez-Valenzuela
reviewed by: Július Korbaš
MaRDI profile type: Publication
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description: scientific article; zbMATH DE number 1052951