Connections and splittings of super-manifolds
type: Item
instance of: scholarly article
title: Connections and splittings of super-manifolds
zbMATH Open document ID: 0812.53029
DOI: 10.1016/0926-2245(94)00011-5
author: Jean-Louis Koszul
published in: Differential Geometry and its Applications
publication date: 1995-05-09T00:00:00Z
review text: Let \(M = (M,{\mathcal O})\) be a smooth or analytic supermanifold of dimension \((m,n)\), where \(M\) is the underlying topological space and \(\mathcal O\) the structure sheaf. If \(\mathcal I\) is the sheaf of ideals in \(\mathcal O\) generated by the odd elements, then \(\text{Gr }{\mathcal O} := \bigoplus^ n_{p = 0} ({\mathcal I}^ p/{\mathcal I}^{p + 1})\) is a sheaf of algebras. A splitting of \(M\) is an isomorphism \(\varphi\) of the supermanifold \(\text{Gr }M := (M,\text{Gr } {\mathcal O})\) onto \(M\) such that \(\text{Gr }\varphi\) is the identity of \(\text{Gr }M\). The main result of the paper states that any linear connection on \(M\) determines a splitting of \(M\). This is obtained as an application of an extensive study of derivations and connections in relation with \(m\)-adic filtrations, within a purely algebraic framework. Some other applications are also discussed, as well as a brief geometric approach to the existence of a splitting.
Mathematics Subject Classification ID: 53C05
Mathematics Subject Classification ID: 58A50
zbMATH DE Number: 609683
zbMATH Keywords: supermanifold
zbMATH Keywords: splitting
zbMATH Keywords: linear connection
zbMATH Keywords: \(m\)-adic filtrations
Wikidata QID: Q115362642
reviewed by: E. Vassiliou
MaRDI profile type: Publication
full work available at URL: 0926-2245(94)00011-5
OpenAlex ID: W2031003803
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description: scientific article; zbMATH DE number 609683