Non-split almost complex and non-split Riemannian supermanifolds

Matthias Kalus

Archivum Mathematicum (2019)

  • Volume: 055, Issue: 4, page 229-238
  • ISSN: 0044-8753

Abstract

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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.

How to cite

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Kalus, Matthias. "Non-split almost complex and non-split Riemannian supermanifolds." Archivum Mathematicum 055.4 (2019): 229-238. <http://eudml.org/doc/294151>.

@article{Kalus2019,
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.},
author = {Kalus, Matthias},
journal = {Archivum Mathematicum},
keywords = {supermanifold; almost complex structure; Riemannian metric; non-split},
language = {eng},
number = {4},
pages = {229-238},
publisher = {Department of Mathematics, Faculty of Science of Masaryk University, Brno},
title = {Non-split almost complex and non-split Riemannian supermanifolds},
url = {http://eudml.org/doc/294151},
volume = {055},
year = {2019},
}

TY - JOUR
AU - Kalus, Matthias
TI - Non-split almost complex and non-split Riemannian supermanifolds
JO - Archivum Mathematicum
PY - 2019
PB - Department of Mathematics, Faculty of Science of Masaryk University, Brno
VL - 055
IS - 4
SP - 229
EP - 238
AB - 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.
LA - eng
KW - supermanifold; almost complex structure; Riemannian metric; non-split
UR - http://eudml.org/doc/294151
ER -

References

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