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Another natural lift of a Kähler submanifold of a quaternionic Kähler manifold to the twistor space. (English) Zbl 1080.53041
Let \(\widetilde M^{4n}\) be a quaternionic Kähler manifold and \(\widetilde {\mathcal Z}\) be the twistor space of \(\widetilde M\). Then the natural projection \(\widetilde\pi:\widetilde {\mathcal Z}\to\widetilde M\) is an \(S^2\)-bundle over \(\widetilde M\). Let \(M\) be an almost Hermitian submanifold of \(\widetilde M\). The authors construct a submanifold \({\mathcal Z}\) of \(\widetilde {\mathcal Z}\) such that the natural projection \(\pi: {\mathcal Z}\to M\) is an \(S^1\)-bundle over \(M\). The main result of the paper is the following theorem: Let \(M^{2m}\) be a \(2m(m\geq 2)\)-dimensional Kähler submanifold of a quaternionic Kähler manifold \(\widetilde M\) of positive scalar curvature. Then \({\mathcal Z}\) is a totally real and minimal submanifold of the twistor space \(\widetilde {\mathcal Z}\). in particular the space \({\mathcal Z}\) of a half dimensional Kähler submanifold \(M^{2n}\) of \(\widetilde M^{4n}\) is a minimal Lagrangian submanifold of \(\widetilde {\mathcal Z}\).

MSC:
53C26 Hyper-Kähler and quaternionic Kähler geometry, “special” geometry
53C40 Global submanifolds
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