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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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