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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
##### Keywords:
quaternionic Kähler manifold; twistor space; natural lift
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##### References:
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