Twistor methods in conformal almost symplectic geometry

Abstract

Given a 2n-dimensional almost symplectic manifold $\left(M,\omega\right)$, we consider the conformal class of $\omega$ and to each symplectic connection, $\nabla$, we associate, in a natural way, a $e^{2\sigma}\omega$-symplectic connection, $\nabla^{\sigma}$. We prove that the twistor bundle $Z\left(M,\omega\right):=\frac{P\left(M,Sp\left(2n\right)\right)}{U(n)}$, with its canonical almost complex structure induced by $\nabla$, is an invariant of the conformal class of $\left(\omega,\nabla\right)$. Then we study the interplay between conformal properties of $\left(M,\omega\right)$ and complex properties of $Z\left(M,\omega\right)$, passing trough the existence of special symplectic connections. Finally we prove that, in the case of a special K$\ddot{\textrm{a}}$hler manifold, the section of $Z\left(M,\omega\right)$ defined by the complex structure of M is an almost complex submanifold with respect to a certain almost complex structure on $Z\left(M,\omega\right)$.