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Real hypersurfaces of type \(B\) in complex two-plane Grassmannians related to the Reeb vector. (English) Zbl 1206.53064
The authors give a new characterization of real hypersurfaces of type \(B\), that is, a tube over a totally geodesic \({\mathbb Q}P^{n}\) in a complex two-plane Grassmannian \(G_{2}({\mathbb C}^{m+2})\), where \(m=2n\), with the Reeb vector \(\xi \) belonging to the distribution \({\mathfrak D}\), where \({\mathfrak D}\) denotes a subdistribution in the tangent space \(T_{x}M\) such that \(T_{x}M={\mathfrak D}\oplus {\mathfrak D}^{\bot }\) for any point \(x\in M\) and \({\mathfrak D}^{\bot }=\text{Span}\{\xi _{1},\xi _{2},\xi _{3}\}\). The main theorem of the paper is as follows:
“Let \(M\) be a connected orientable Hopf real hypersurface in complex two-plane Grassmannians \(G_{2}({\mathbb C}^{m+2})\), \(m\geq 3\). Then the Reeb vector \(\xi \) belongs to the distribution \({\mathfrak D}\) if and only if \(M\) is locally congruent to an open part of a tube around a totally geodesic \({\mathbb Q}P^{n}\) in \(G_{2}({\mathbb C}^{m+2})\), where \(m=2n\).”

MSC:
53C40 Global submanifolds
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
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