On the twistor spaces of almost Hermitian manifolds (Q1264246)

Given an almost Hermitian manifold \((M,g,J)\), denote by \({\mathcal G}_k(M)\) the Grassmann bundle of \(J\)-invariant \(2k\)-subspaces of \(TM\). Any unitary connection \(\omega\) on \(M\) gives rise to a splitting of \(T{\mathcal G}_k(M)\) into vertical and horizontal parts which allows one to introduce four almost complex structures \(J_{\pm}\) and \(F_{\pm}\) on \({\mathcal G}_k(M)\). The structures \(J_{\pm}\), defined in a way standard for the twistor theory, were studied by \textit{N. R. O'Brian} and \textit{J. H. Rawnsley} [Ann. Global Anal. Geom. 3, 29-58 (1985; Zbl 0526.53057)]. As to \(F_{\pm}\), on the vertical subspace of \(T_{w}{\mathcal G}_k(M)\), \(F_{\pm}=\pm\) the complex structure of the fibre through \(w\) (\(=J_{\pm}\)) while on the horizontal subspace \(F_{\pm}=\) the horizontal lift of \(J\). The author proves that in the case when \(\omega\) is the Hermitian (\(\equiv\) Chern, or canonical) connection of \(M\), \(F_{\pm}\) is integrable if and only if \(J\) is integrable. Similar to the usual twistor space \({\mathcal Z}(M)\) of \(M\), the generalized twistor space \({\mathcal G}_k(M)\) admits a natural family of Riemannian metrics \(h_r, r>0\), defined by means of \(\omega\). These metrics are compatible with \(F_{\pm}\) and \(J_{\pm}\), and the author shows that \((h_r,F_{\pm})\) is a Kähler structure on \({\mathcal G}_k(M)\) if and only if \(M\) is a flat Kähler manifold provided the almost complex structure \(J\) is integrable and \(\omega\) is the Hermitian connection. Under the latter assumptions, \((h_r,J_{\pm})\) is an almost Kähler structure on the usual twistor space \({\mathcal Z}(M)\) of a 4-manifold \(M\) if and only if \(M\) is Kähler-Einstein with scalar curvature \(\pm 2/r^2\). This supplements results obtained earlier by \textit{O. Mushkarov} [C. R. Acad. Sci., Sér. I 305, 307-309 (1987; Zbl 0626.53028)], \textit{G. R. Jensen} and \textit{M. Rigoli} [Contemp. Math. 101, 197-232 (1989; Zbl 0688.53026)], \textit{P. Gauduchon} [Ann. Sc. Norm. Supér. Pisa, Cl. Sci., 563-629 (1991; Zbl 0763.53034)]. In the case of a 4-dimensional manifold \(M\) with a unitary connection on it, the author gives local formulas for the Hermitian connection of \((h_r,F_{\pm})\) and \((h_r,J_{\pm})\) on the twistor space \({\mathcal Z}(M)\) and obtains formulas for the corresponding curvature forms (representing the first Chern class). These fit to some formulas for the first Chern class of \(J_{\pm}\) and \(F_{\pm}\) established in different cases by \textit{P. Gauduchon} [op. cit.], \textit{T. Davidov, O. Mushkarov} and \textit{G. Grantcharov} [in: Proceedings of the International Workshop on Almost Complex Structures, Sofia (1992), 113-149 (1994; Zbl 0865.53024)], \textit{V. V. Tsanov} [in: Differential geometric methods in theoretical physics (Clausthal, FRG, 1986), 507-517 (1987; Zbl 0713.32011)].