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Forum Mathematicum

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Ed. by Cohen, Frederick R. / Droste, Manfred / Darmon, Henri / Duzaar, Frank / Echterhoff, Siegfried / Gordina, Maria / Neeb, Karl-Hermann / Shahidi, Freydoon / Sogge, Christopher D. / Takayama, Shigeharu / Wienhard, Anna

IMPACT FACTOR 2018: 0.867

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Volume 30, Issue 3


Parabolic conformally symplectic structures I; definition and distinguished connections

Andreas ČapORCID iD: http://orcid.org/0000-0002-7745-3708 / Tomáš SalačORCID iD: http://orcid.org/0000-0001-5517-4196
Published Online: 2017-10-03 | DOI: https://doi.org/10.1515/forum-2017-0018


We introduce a class of first order G-structures, each of which has an underlying almost conformally symplectic structure. There is one such structure for each real simple Lie algebra which is not of type Cn and admits a contact grading. We show that a structure of each of these types on a smooth manifold M determines a canonical compatible linear connection on the tangent bundle TM. This connection is characterized by a normalization condition on its torsion. The algebraic background for this result is proved using Kostant’s theorem on Lie algebra cohomology. For each type, we give an explicit description of both the geometric structure and the normalization condition. In particular, the torsion of the canonical connection naturally splits into two components, one of which is exactly the obstruction to the underlying structure being conformally symplectic. This article is the first in a series aiming at a construction of differential complexes naturally associated to these geometric structures.

Keywords: Almost conformally symplectic structure; special symplectic connection; exceptional symplectic holonomy; finite order geometric structure; first prolongation; canonical connection

MSC 2010: : 53D15; 53B15; 53C10; 53C15; : 53C29; 53C55


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About the article

Received: 2017-01-27

Revised: 2017-09-05

Published Online: 2017-10-03

Published in Print: 2018-05-01

Funding Source: Austrian Science Fund

Award identifier / Grant number: P23244-N13

Award identifier / Grant number: P27072-N25

Support by projects P23244-N13 (both authors) and P27072-N25 (first author) of the Austrian Science Fund (FWF) is gratefully acknowledged.

Citation Information: Forum Mathematicum, Volume 30, Issue 3, Pages 733–751, ISSN (Online) 1435-5337, ISSN (Print) 0933-7741, DOI: https://doi.org/10.1515/forum-2017-0018.

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