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# Forum Mathematicum

Managing Editor: Bruinier, Jan Hendrik

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

CiteScore 2018: 0.71

SCImago Journal Rank (SJR) 2018: 0.898
Source Normalized Impact per Paper (SNIP) 2018: 0.964

Mathematical Citation Quotient (MCQ) 2018: 0.71

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1435-5337
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Volume 30, Issue 3

# Parabolic conformally symplectic structures I; definition and distinguished connections

Andreas Čap
/ Tomáš Salač
Published Online: 2017-10-03 | DOI: https://doi.org/10.1515/forum-2017-0018

## Abstract

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 ${C}_{n}$ 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 $\mathrm{TM}$. 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.

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

## References

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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,

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