On Twistor Almost Complex Structures

Abstract

In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J_{\nabla }^{\pm }$ defined on the twistor space $J(M,g)$ of an even-dimensional manifold $M$ with additional structures $g$ and $\nabla$ a $g$-connection. We also look at the question of the compatibility of $J_{\nabla }^{\pm }$ with a natural closed $2$-form ${\omega }^{J(M,g,\nabla )}$ defined on $J(M,g)$. For $(M,g)$ we consider either a pseudo-Riemannian manifold, orientable or not, with the Levi Civita connection or a symplectic manifold with a given symplectic connection $\nabla$. In all cases $J(M,g)$ is a bundle of complex structures on the tangent spaces of $M$ compatible with $g$ and we denote by $\pi :J(M,g)⟶M$ the bundle projection. In the case $M$ is oriented we require the orientation of the complex structures to be the given one. In the symplectic case the complex structures are positive.

21 pages; fixed some typos and re-laid some formulas for easier reading