Address: Fakultät für Mathematik, Ruhr-Universität Bochum, D-44780 Bochum, Germany
E-mail: matthias.kalus@rub.de
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.
AMSclassification: primary 32Q60; secondary 53C20, 58A50.
Keywords: supermanifold, almost complex structure, Riemannian metric, non-split.
DOI: 10.5817/AM2019-4-229