# American Institute of Mathematical Sciences

doi: 10.3934/jgm.2021006

## On twistor almost complex structures

 1 Membres de l'Académie Royale de Belgique, Université Libre de Bruxelles, Département de Mathématique, Campus Plaine, CP 218, boulevard du triomphe, BE-1050 Bruxelles, Belgium 2 Mathematics Institute, Zeeman Building, University of Warwick, Coventry CV4 7AL, United Kingdom

* Corresponding author: Simone Gutt

Dedicated to our friend Kirill Mackenzie

Received  November 2020 Published  April 2021

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 measure their non-integrability by the dimension of the span of the values of $N^{J^\pm_\nabla}$. 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 $g$. 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.

Citation: Michel Cahen, Simone Gutt, John Rawnsley. On twistor almost complex structures. Journal of Geometric Mechanics, doi: 10.3934/jgm.2021006
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