Symplectic connections

  title={Symplectic connections},
  author={Pierre Bieliavsky and Michel Cahen and Simone Gutt and John Howard Rawnsley and Lorenz J. Schwachhofer},
This article is an overview of the results obtained in recent years on symplectic connections. We present what is known about preferred connections (critical points of a variational principle). The class of Ricci-type connections (for which the curvature is entirely determined by the Ricci tensor) is described in detail, as well as its far reaching generalization to special connections. A twistorial construction shows a relation between Ricci-type connections and complex geometry. We give a… 

Remarks on Symplectic Connections

This note contains a short survey on some recent work on symplectic connections: properties and models for symplectic connections whose curvature is determined by the Ricci tensor, and a procedure to

Remarks on symplectic sectional curvature

Connections on contact manifolds and contact twistor space

In this paper we generalize the definition of symplectic connection to the contact case. It turns out that any odd-dimensional manifold equipped with a contact form admits contact connections and

Symplectic connections induced by the Chern connection

Let (M;!) be a symplectic manifold and F be a Finsler struc- ture on M. In the present paper we deflne a lift of the symplectic two-form ! on the manifold TMn0, and flnd the conditions that the Chern

Critical symplectic connections on surfaces

The space of symplectic connections on a symplectic manifold is a symplectic affine space. M. Cahen and S. Gutt showed that the action of the group of Hamiltonian diffeomorphisms on this space is

Remarks on symplectic twistor spaces

We consider some classical fibre bundles furnished with almost complex structures of twistor type, deduce their integrability in some cases and study self-holomorphic sections of the general twistor

Symmetries of the Space of Linear Symplectic Connections

There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment

Extremal symplectic connections on surfaces

M. Cahen and S. Gutt found the moment map for the action of the symplectomorphism group on the symplectic affine space of symplectic connections on a symplectic manifold. This paper studies extremal

On a Connection Used in Deformation Quantization

Using natural lifting operations, we give a coordinate-free proof of the fact that the connection used by Bordemann, Neumaier and Waldmann to construct the Fedosov standard ordered star product on




  • C. Boubel
  • Mathematics
    Proceedings of the Edinburgh Mathematical Society
  • 2003
Abstract A symplectic connection on a symplectic manifold, unlike the Levi-Civita connection on a Riemannian manifold, is not unique. However, some spaces admit a canonical connection (symmetric

Symplectic curvature tensors

In the paper, one establishes the decomposition of the space of tensors which have the symmetries of the curvature of a torsionless symplectic connection into Sp (n)-irreducible components. This

Reduction, induction and Ricci flat symplectic connections

In this paper we present a construction of Ricci-flat connections through an induction procedure. Given a symplectic manifold (M, ω) of dimension 2n, we define induction as a way to construct a

A Construction of Symplectic Connections Through Reduction

We give an elementary construction of symplectic connections through reduction. This provides an elegant description of a class of symmetric spaces and gives examples of symplectic connections with

Construction of Ricci-type connections by reduction and induction

Given the Euclidean space ℝ2n+2 endowed with a constant symplectic structure and the standard flat connection, and given a polynomial of degree 2 on that space, Baguis and Cahen [1] have defined a

A variational principle for symplectic connections

Symplectic connections with parallel Ricci tensor

A variational principle introduced to select some symplectic connections lead- s to Þeld equations which, in the case of the Levi Civita connection of Kahler manifolds, are equivalent to the

Symplectic twistor spaces

Homogeneous connections with special symplectic holonomy

Abstract. We classify all homogeneous symplectic manifolds with a torsion free connection of special symplectic holonomy, i.e. a connection whose holonomy is an absolutely irreducible proper subgroup