On twistor almost complex structures

<p style='text-indent:20px;'>In this paper we look at the question of integrability, or not, two natural almost complex structures <inline-formula><tex-math id="M1">\begin{document}$ J^{\pm}_\nabla $\end{document}</tex-math></inline-formula> defined on twistor space id="M2">\begin{document}$ J(M, g) an even-dimensional manifold id="M3">\begin{document}$ M with additional id="M4">\begin{document}$ g and id="M5">\begin{document}$ \nabla a id="M6">\begin{document}$ $\end{document}</tex-math></inline-formula>-connection. We measure their non-integrability by dimension span values id="M7">\begin{document}$ N^{J^\pm_\nabla} $\end{document}</tex-math></inline-formula>. also compatibility id="M8">\begin{document}$ closed id="M9">\begin{document}$ 2 $\end{document}</tex-math></inline-formula>-form id="M10">\begin{document}$ \omega^{J(M, g, \nabla)} id="M11">\begin{document}$ For id="M12">\begin{document}$ (M, consider either pseudo-Riemannian manifold, orientable Levi Civita connection symplectic given id="M13">\begin{document}$ In all cases id="M14">\begin{document}$ is bundle tangent spaces id="M15">\begin{document}$ compatible id="M16">\begin{document}$ case id="M17">\begin{document}$ oriented require orientation to be one. are positive.</p>