Connections and splittings of super-manifolds
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.
scientific article; zbMATH DE number 609683
scientific article