On twistor almost complex structures

Abstract

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