Holonomy of supermanifolds

Abstract

Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces are obtained. As the special case, the holonomy of linear connections on supermanifolds is studied. Examples of parallel geometric structures on supermanifolds and the corresponding holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger superalgebras are defined and their examples are given.