Monatshefte für Mathematik

, Volume 171, Issue 3–4, pp 357–376 | Cite as

Real hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster Reeb parallel shape operator

  • Imsoon Jeong
  • Makoto Kimura
  • Hyunjin Lee
  • Young Jin Suh
Article

Abstract

In a paper due to Jeong et al. (Kodai Math J 34(3):352–366, 2011) we have shown that there does not exist a hypersurface in \(G_{2}({\mathbb{C }}^{m+2})\) with parallel shape operator in the generalized Tanaka–Webster connection (see Tanaka in Jpn J Math 20:131–190, 1976; Tanno in Trans Am Math Soc 314(1):349–379, 1989). In this paper, we introduce the notion of the Reeb parallel 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 \(G_2(\mathbb{C }^{m+2})\).

Keywords

Real hypersurfaces Complex two-plane Grassmannians   Hopf hypersurface Generalized Tanaka–Webster connection  Reeb parallel shape operator \(\mathcal F \)-parallel shape operator 

Mathematics Subject Classification (2000)

53C40 53C15 

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Copyright information

© Springer-Verlag Wien 2013

Authors and Affiliations

  • Imsoon Jeong
    • 1
  • Makoto Kimura
    • 2
  • Hyunjin Lee
    • 3
  • Young Jin Suh
    • 1
  1. 1.Department of MathematicsKyungpook National UniversityTaeguKorea
  2. 2.Department of MathematicsIbaraki UniversityMitoJapan
  3. 3.Graduate School of Electrical Engineering and Computer ScienceKyungpook National UniversityTaeguKorea

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