Non-split almost complex and non-split Riemannian supermanifolds

Abstract

Non-split almost complex supermanifolds and non-split Riemannian supermanifolds are studied. The first obstacle for a splitting is parametrized by group orbits on an infinite dimensional vector space. For almost complex structures, the existence of a splitting is equivalent to the existence of local coordinates in which the almost complex structure can be represented by a purely numerical matrix, i.e. containing no Grassmann variables. For Riemannian metrics, terms up to degree 2 are allowed in such a local matrix representation, in order to preserve non-degeneracy. It is further shown that non-split structures appear in the almost complex case as deformations of a split reduction and in the Riemannian case as the deformation of an underlying metric. In contrast to non-split deformations of complex supermanifolds, these deformations can be restricted by cut-off functions to local deformations. A class of examples of nowhere split structures constructed from almost complex manifolds of dimension $6$ and higher, is provided for both cases.