Twistor Bundles of Almost Symplectic Manifolds

Abstract

In this paper we introduce the twistor bundle of a 2n-dimensional almost symplectic manifold M as the quotient bundle $\frac{P\left(M,Sp\left(2n\right)\right)}{U\left(n\right)}$. Given a symplectic connection on M we introduce a natural almost Hermitian structure on the twistor bundle and we prove that this structure is K$\ddot{\textrm{a}}$hler if and only if M is symplectic and the chosen connection has vanishing curvature and (0,2)-part of the torsion. Moreover we prove that in the case of $\mathbb{R}^{2n}$ with standard symplectic structure the twistor bundle turns out to be K$\ddot{\textrm{a}}$hler with constant scalar curvature for a certain class of symplectic connections.