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Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition. (English) Zbl 1265.53075
The authors give a description of Hopf hypersurfaces $$M$$ in the Grassmannian $$N= \text{Gr}_2(\mathbb {C}^n)$$ of 2-planes in the complex space whose shape operator $$A$$ satisfies some commuting condition. Recall that a hypersurface $$M$$ with the unit normal $$n$$ in a Kähler manifold is called a Hopf hypersurface if the Reeb vector $$\xi := Jn$$ is an eigenvector of the shape operator $$A$$. The Grassmann manifold $$G$$ has a natural Kähler structure $$(g,J)$$ and a natural quaternionic Kähler structure $$(g,Q)$$ where the (parallel) quaternionic structure is locally generated by three locally defined almost complex structures $$J_1,J_2,J_3$$ which commute with the complex structure $$J$$. Denote by $$\varphi , \varphi _i, \, i =1,2,3$$ the endomorphisms of $$TM$$ induces by the endomorphisms $$J, J_i \in \text{End}(TN)_M= \text{End}( \mathbb {R}n \oplus TM)$$. The authors prove that an oriented Hopf hypersurface $$M \subset \text{Gr}_2(\mathbb {C}^n)$$ such that $\varphi \circ \varphi _1 \circ A = A \circ \varphi \circ \varphi _1$ is an open part of the tube around a totally geodesic submanifold $$\text{Gr}_2(\mathbb {C}^{n-1})$$ of $$\text{Gr}_2(\mathbb {C}^n)$$.

##### MSC:
 53C50 Global differential geometry of Lorentz manifolds, manifolds with indefinite metrics 53C55 Global differential geometry of Hermitian and Kählerian manifolds 53C40 Global submanifolds 11R52 Quaternion and other division algebras: arithmetic, zeta functions
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##### References:
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