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Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster Reeb parallel shape operator. (English) Zbl 1277.53049
Summary: In a paper due to the first author et al. [Kodai Math. J. 34, No. 3, 352–366 (2011; Zbl 1231.53049)] we have shown that there does not exist a hypersurface in the space $$G_{2}({\mathbb{C}}^{m+2})$$ of all complex 2-dimensional linear subspaces of $$\mathbb{C}^{m+2}$$ with parallel shape operator in the generalized Tanaka-Webster connection (see [N. Tanaka, Jap. J. Math., New Ser. 2, 131–190 (1976; Zbl 0346.32010); S. Tanno, Trans. Am. Math. Soc. 314, No. 1, 349–379 (1989; Zbl 0677.53043)]). In this paper, we introduce the notion of Reeb parallelism in the sense of generalized Tanaka-Webster connection for a hypersurface $$M$$ in $$G_{2}({\mathbb{C }}^{m+2})$$ and prove that $$M$$ is an open part of a tube around a totally geodesic $$G_2(\mathbb{C }^{m+1})$$ in $$G_2(\mathbb{C }^{m+2})$$.

##### MSC:
 53C40 Global submanifolds 53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.) 53C30 Differential geometry of homogeneous manifolds
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##### References:
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