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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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