Non-split almost complex and non-split Riemannian supermanifolds

TL;DR

This paper investigates non-split structures in almost complex and Riemannian supermanifolds, showing how they arise as deformations and providing examples, with implications for understanding their local and global properties.

Contribution

It introduces a new perspective on non-split supermanifolds by analyzing their deformation theory and constructing explicit examples in higher dimensions.

Findings

Non-split structures are characterized by group orbits on infinite-dimensional spaces.

Deformations of non-split structures can be localized using cut-off functions.

Explicit examples of nowhere split structures are constructed from higher-dimensional almost complex manifolds.

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. Further it is shown that non-split structures appear in the first case as deformations of a split reduction and in the second 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.