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Symplectic twistor spaces. (English) Zbl 0629.53032
Let (M,$$\omega)$$ be an almost symplectic manifold and $${\mathcal T}(M,\omega)$$ its symplectic twistor space, i.e. the bundle of the compatible complex structures of its tangent spaces, as considered by M. Dubois-Violette [Mathématique et physique, Sémin. Éc. Norm. Supér., Paris 1979-1982, Prog. Math. 37, 1-42 (1983; Zbl 0522.53029)], L. Bérard-Bergery and T. Ochiai [Global Riemannian geometry, Proc. Symp., Durham/Engl. 1982, 52-59 (1984) and ibid. 85-92 (1984; Zbl 0615.53036)] and N. R. O’Brian and J. H. Rawnsley [Ann. Global Anal. Geom. 3, 29-58 (1985; Zbl 0526.53057)]. The main results in the paper show that $${\mathcal T}(M,\omega)$$ has an either integrable or almost Kähler structure if and only if M is locally symplectic flat. Applications are given to the study of the differential forms on M and of the mappings $$\phi$$ : $$N\to M$$, where N is a Kähler manifold.
Reviewer: J.Weinstein

##### MSC:
 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C55 Global differential geometry of Hermitian and Kählerian manifolds 37J99 Dynamical aspects of finite-dimensional Hamiltonian and Lagrangian systems
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