Special Symplectic Connections
Michel Cahen,Lorenz Schwachhöfer +1 more
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In this paper, it was shown that the symplectic reduction of a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way and that any special manifold or orbifold with such a connection is locally equivalent to one of these symplectic reductions.Abstract:
By a special symplectic connection we mean a torsion free connection which is either the Levi-Civita connection of a Bochner-Kahler metric of arbitrary signature, a Bochner-bi-Lagrangian connection, a connection of Ricci type or a connection with special symplectic holonomy. A manifold or orbifold with such a connection is called special symplectic. We show that the symplectic reduction of (an open cell of) a parabolic contact manifold by a symmetry vector field is special symplectic in a canonical way. Moreover, we show that any special symplectic manifold or orbifold is locally equivalent to one of these symplectic reductions. As a consequence, we are able to prove a number of global properties, including a classification in the compact simply connected case.read more
Citations
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Parabolic conformally symplectic structures I; definition and distinguished connections
Andreas Cap,Tomas Salac +1 more
TL;DR: In this article, a class of first order G-structures, each of which has an underlying almost conformally symplectic structure, is introduced, and it is shown that a structure of each of these types on a smooth manifold determines a canonical compatible linear connection on the tangent bundle.
Journal ArticleDOI
Pushing down the Rumin complex to conformally symplectic quotients
Andreas Cap,Tomas Salac +1 more
TL;DR: In this article, it was shown that any local leaf space M for the foliation determined by ξ naturally carries a conformally symplectic (cs-) structure, and the Rumin complex on M # descends to a complex of differential operators on M, whose cohomology can be computed.
Posted ContentDOI
Parabolic conformally symplectic structures III; Invariant differential operators and complexes
Andreas Cap,Tomas Salac +1 more
TL;DR: In this article, a family of geometric structures (PACS-structures) which all have an underlying almost conformally symplectic structure was studied, and it was shown that all connections of exotic symplectic holonomy arise as the canonical connection of such a structure.
Journal ArticleDOI
Parabolic symmetric spaces
TL;DR: In this paper, a smooth system of symmetries on |1|-graded parabolic geometries was studied and the existence of an invariant admissible affine connection under a weak condition on the system was shown.
Journal ArticleDOI
Parabolic conformally symplectic structures II: parabolic contactification
Andreas Cap,Tomas Salac +1 more
TL;DR: In this article, a parabolic contact structure with a transversal infinitesimal automorphism is constructed on any local leaf space of the corresponding foliation, and it is shown that any PCS-structure can be locally realized (uniquely up to isomorphism).
References
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Book
Introduction to Lie Algebras and Representation Theory
TL;DR: In this paper, Semisimple Lie Algebras and root systems are used for representation theory, isomorphism and conjugacy theorem, and existence theorem for representation.
Book
Curvature and Betti numbers
健太郎 矢野,Salomon Bochner +1 more
TL;DR: In this paper, the authors proposed a pseudo-harmonic tensors and pseudo-killing tensors in metric Manifolds with Torsion, which can be seen as a kind of semi-simple group spaces.
Journal ArticleDOI
Classification of irreducible holonomies of torsion-free affine connections
TL;DR: The subgroups of GL(n,R) that act irreducibly on R^n and that can occur as the holonomy of a torsion-free affine connection on an n-manifold are classified in this paper.
Journal ArticleDOI
A variational principle for symplectic connections
Frédéric Bourgeois,Michel Cahen +1 more
TL;DR: In this paper, a variational principle for simply connected simply connected symplectic manifolds is introduced and the corresponding field equations are studied for two-dimensional compact and non-compact simply connected manifold.