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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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