On Twistor Almost Complex Structures

TL;DR

This paper investigates the integrability and compatibility of natural almost complex structures on twistor spaces of even-dimensional manifolds with additional geometric structures, exploring conditions under which these structures are integrable or compatible with natural forms.

Contribution

It provides new insights into the conditions for integrability and compatibility of twistor almost complex structures on various geometric manifolds.

Findings

Conditions for integrability of $J^{ abla}_ abla$ structures.

Criteria for compatibility with the natural 2-form.

Analysis of structures on pseudo-Riemannian and symplectic manifolds.

Abstract

In this paper we look at the question of integrability, or not, of the two natural almost complex structures $J^{\pm}_\nabla$ 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 $J^{\pm}_\nabla$ 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 $\pi \colon J(M,g) \longrightarrow M$ 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.