Twistor space of a generalized quaternionic Kähler manifold
A quaternionic manifold is defined by a \(G\)-structure admitting a torsion-free connection, where \(G\) denotes the maximal subgroup \(\mathrm{GL}(n,\mathbb H)\cdot\mathrm{GL}(1, \mathbb H)\) of \(\mathrm{GL}(4n, \mathbb R)\). Similarly it can be defined as a subbundle \(H\) of the endomorphism bundle \(\mathcal{E}nd (TM)\) such that each fiber \(H_x\) is isomorphic (as a real algebra) to the quaternion algebra \(\mathbb{H}\). The subbundle \(H\) is called the almost quaternionic structure bundle. A manifold equipped with an almost quaternionic structure is called an almost quaternionic manifold. In the author's words: ``Specifying an almost quaternionic structure \(Q\) on a \(4n\)-manifold \(M\) is equivalent to specifying a twistor bundle \(Z(Q)\longrightarrow M\). When \(Q\) is invariant under a torsion-free connection, \(Z(Q)\) can be endowed with an almost complex structure \(J\). For \(n = 1\) \textit{M. F. Atiyah} et al. [Proc. R. Soc. Lond., Ser. A 362, 425--461 (1978; Zbl 0389.53011)] have related the integrability of \(J\) to the geometry of \((M,Q)\). For \(n > 1\) \textit{S. Salamon} [Invent. Math. 67, 143--171 (1982; Zbl 0486.53048); in: Metriche invarianti, applicazioni armoniche e questioni connesse, Conv. 1981. Symp. Math. 26, 139--151 (1982; Zbl 0534.53030)] showed that the almost complex structure \(J\) on \(Z(Q)\) is always integrable. Recently, \textit{R. Pantilie} [Ann. Mat. Pura Appl. (4) 193, No. 3, 633--641 (2014; Zbl 1312.53105)] introduced the concept of a generalized quaternionic Kähler structure \(Q\) on \(M\), defined a generalized twistor space \(Z(Q)\), and showed that \(Z(Q)\) comes naturally equiped with a tautological almost generalized complex structure, but leaves open the problem of the integrability.'' The author fills precisely this gap, obtaining the conditions needed by a generalized quaternionic Kähler structure to be integrable. He includes a good bibliography and examples. A sound paper on quaternionic manifolds is [\textit{S. M. Salamon}, Ann. Sci. Éc. Norm. Supér. (4) 19, No. 1, 31--55 (1986; Zbl 0616.53023)].
scientific article; zbMATH DE number 6812743
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