Symplectic Connections

Abstract

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 construction of Ricci-flat symplectic connections. We end up by presenting, through an explicit example, an approach to noncommutative symplectic symmetric spaces.math.SG/0511194 v2, May 2006. Section 6.8 rewritten.

Remarks on symplectic sectional curvature

“…The reader is referred to the references mentioned in the introduction and the survey [4] for background.…”

“…Among the many available references some basic ones are [1], [10], and [11], in addition to [19]. The survey [4] is a good starting point and contains further references.…”

“…Since tr ad(x) = 2x 1 , g is not unimodular and ℓ = −2e 4 . Calculations shows that [s(x), s(y)] = Since s(x)ℓ is in the isotropic subspace span{e 3 , e 4 }, it follows from (8.11) that Ω(R(x, y)ℓ, ℓ) = 0 for all x, y ∈ g. In particular, the symplectic sectional curvature of any symplectic subspace of g containing ℓ is not definite.…”

Second-Order Deformations of Hydrodynamic-Type Poisson Brackets

“…a flat symplectic connection (see, for example, [2]). In [7] such a pair (ω, ∇) was referred to as a Fedosov structure.…”

Symplectic Yang–Mills theory, Ricci tensor, and connections

“…In recent years, this problem has been approached by studying symplectic connections that, in addition to torsion-freeness, satisfy suitable curvature conditions. For an overview, see [2] and the references therein. In [3], the authors derived such a condition by a variational principle using a Lagrangian density with a quadratic polynomial in the curvature.…”