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Parabolic conformally symplectic structures I; definition and distinguished connections

  • Andreas Čap ORCID logo EMAIL logo and Tomáš Salač ORCID logo
Published/Copyright: October 3, 2017
Forum Mathematicum
From the journal Forum Mathematicum Volume 30 Issue 3

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 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

Communicated by Anna Wienhard

Funding source: Austrian Science Fund

Award Identifier / Grant number: P23244-N13

Award Identifier / Grant number: P27072-N25

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

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Received: 2017-1-27
Revised: 2017-9-5
Published Online: 2017-10-3
Published in Print: 2018-5-1

© 2018 Walter de Gruyter GmbH, Berlin/Boston

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
https://doi.org/10.1515/forum-2017-0018
Keywords for this article
Almost conformally symplectic structure; special symplectic connection; exceptional symplectic holonomy; finite order geometric structure; first prolongation; canonical connection

Articles in the same Issue

  1. Frontmatter
  2. A generalized uniqueness theorem and the graded ideal structure of Steinberg algebras
  3. Positive solutions for nonlinear nonhomogeneous parametric Robin problems
  4. Period relations for cusp forms of GSp4
  5. A generalization of a graph theory Mertens’ theorem: Galois covering case
  6. Commutators of the fractional integrals for second-order elliptic operators on Morrey spaces
  7. Vojta’s conjecture on rational surfaces and the abc conjecture
  8. The least unramified prime which does not split completely
  9. The ω-inequality problem for concatenation hierarchies of star-free languages
  10. Higher weight on GL(3). I: The Eisenstein series
  11. Fourier transforms of powers of well-behaved 2D real analytic functions
  12. Parabolic conformally symplectic structures I; definition and distinguished connections
  13. The quasi-arithmetic means and Cartan barycenters of compactly supported measures
  14. Estimates of lattice points in the discriminant aspect over abelian extension fields
  15. On Hecke eigenvalues of Siegel modular forms in the Maass space
  16. Harmonicity and minimality of complex and quaternionic radial foliations
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