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Holonomy of supermanifolds

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Abstract

Holonomy groups and holonomy algebras for connections on locally free sheaves over supermanifolds are introduced. A one-to-one correspondence between parallel sections and holonomy-invariant vectors, and a one-to-one correspondence between parallel locally direct subsheaves and holonomy-invariant vector supersubspaces are obtained. As the special case, the holonomy of linear connections on supermanifolds is studied. Examples of parallel geometric structures on supermanifolds and the corresponding holonomies are given. For Riemannian supermanifolds an analog of the Wu theorem is proved. Berger superalgebras are defined and their examples are given.

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Authors and Affiliations

  1. Department of Algebra and Geometry, Masaryk University in Brno, Kotlářská 2, 611 37, Brno, Czech Republic

    Anton S. Galaev

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  1. Anton S. Galaev
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Correspondence to Anton S. Galaev.

Additional information

Communicated by V. Cortés.

Supported from the Basic Research Center no. LC505 (Eduard Čech Center for Algebra and Geometry) of Ministry of Education, Youth and Sport of Czech Republic.

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Galaev, A.S. Holonomy of supermanifolds. Abh. Math. Semin. Univ. Hambg. 79, 47–78 (2009). https://doi.org/10.1007/s12188-008-0015-7

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  • DOI: https://doi.org/10.1007/s12188-008-0015-7

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