# Non-split almost complex and non-split Riemannian supermanifolds

Archivum Mathematicum (2019)

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

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topKalus, 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 -

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