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Advances in Geometry

Managing Editor: Grundhöfer, Theo / Joswig, Michael

Editorial Board: Bamberg, John / Bannai, Eiichi / Cavalieri, Renzo / Coskun, Izzet / Duzaar, Frank / Eberlein, Patrick / Gentili, Graziano / Henk, Martin / Kantor, William M. / Korchmaros, Gabor / Kreuzer, Alexander / Lagarias, Jeffrey C. / Leistner, Thomas / Löwen, Rainer / Ono, Kaoru / Ratcliffe, John G. / Scheiderer, Claus / Van Maldeghem, Hendrik / Weintraub, Steven H. / Weiss, Richard

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𝔇-parallel normal Jacobi operators for Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka–Webster connection

Eunmi Pak / Young Jin Suh
Published Online: 2019-07-04 | DOI: https://doi.org/10.1515/advgeom-2019-0012


We study classifying problems for real hypersurfaces in a complex two-plane Grassmannian G2(ℂm+2). In relation to the generalized Tanaka–Webster connection, we consider a new concept of parallel normal Jacobi operator for real hypersurfaces in G2(ℂm+2) and prove that a real hypersurface in G2(ℂm+2) with generalized Tanaka–Webster 𝔇-parallel normal Jacobi operator is locally congruent to an open part of a tube around a totally geodesic quaternionic projective space ℍPn in G2(ℂm+2), where m = 2n.

Keywords: Real hypersurface; complex two-plane Grassmannian; Hopf hypersurface; generalized Tanaka–Webster connection; normal Jacobi operator; generalized Tanaka–Webster parallel normal Jacobi operator

MSC 2010: Primary 53C40; Secondary 53C15


  • [1]

    D. V. Alekseevskiĭ, Compact quaternion spaces. (Russian) Funkcional. Anal. i Priložen 2 (1968), 11–20. English translation: Functional Analysis Appl. 2 (1968), 106–114. MR0231314 Zbl 0175.19001Google Scholar

  • [2]

    J. Berndt, Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127 (1999), 1–14. MR1666307 Zbl 0920.53016CrossrefGoogle Scholar

  • [3]

    J. Berndt, Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex two-plane Grassmannians. Monatsh. Math. 137 (2002), 87–98. MR1937621 Zbl 1015.53034CrossrefGoogle Scholar

  • [4]

    J. T. Cho, CR structures on real hypersurfaces of a complex space form. Publ. Math. Debrecen 54 (1999), 473–487. MR1694456 Zbl 0929.53029Google Scholar

  • [5]

    J. de Dios Pérez, I. Jeong, Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with commuting normal Jacobi operator. Acta Math. Hungar. 117 (2007), 201–217. MR2361601 Zbl 1220.53070CrossrefGoogle Scholar

  • [6]

    J. de Dios Pérez, Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying ∇Ui R = 0. Differential Geom. Appl. 7 (1997), 211–217. MR1480534 Zbl 0901.53011CrossrefGoogle Scholar

  • [7]

    I. Jeong, H. J. Kim, Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator. Publ. Math. Debrecen 76 (2010), 203–218. MR2598182 Zbl 1274.53080Google Scholar

  • [8]

    I. Jeong, Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with 𝔉-parallel normal Jacobi operator. Kyungpook Math. J. 51 (2011), 395–410. MR2874974 Zbl 1243.53099CrossrefGoogle Scholar

  • [9]

    U.-H. Ki, J. de Dios Pérez, F. G. Santos, Y. J. Suh, Real hypersurfaces in complex space forms with ξ-parallel Ricci tensor and structure Jacobi operator. J. Korean Math. Soc. 44 (2007), 307–326. MR2295391 Zbl 1144.53069CrossrefWeb of ScienceGoogle Scholar

  • [10]

    M. Kon, Real hypersurfaces in complex space forms and the generalized Tanaka-Webster connection. In: Proceedings of the 13th International Workshop on Differential Geometry and Related Fields, 145–159, Natl. Inst. Math. Sci., Taejŏn 2009. MR2641131 Zbl 1185.53058Google Scholar

  • [11]

    H. Lee, Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassmannians related to the Reeb vector. Bull. Korean Math. Soc. 47 (2010), 551–561. MR2666376 Zbl 1206.53064Web of ScienceCrossrefGoogle Scholar

  • [12]

    E. Pak, J. de Dios Pérez, C. J. G. Machado, C. Woo, Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster parallel normal Jacobi operator. Czechoslovak Math. J. 65 (140) (2015), 207–218. MR3336034 Zbl 1363.53049Web of ScienceGoogle Scholar

  • [13]

    N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections. Japan. J. Math. (N.S.) 2 (1976), 131–190. MR0589931 Zbl 0346.32010CrossrefGoogle Scholar

  • [14]

    S. Tanno, Variational problems on contact Riemannian manifolds. Trans. Amer. Math. Soc. 314 (1989), 349–379. MR1000553 Zbl 0677.53043CrossrefGoogle Scholar

  • [15]

    S. M. Webster, Pseudo-Hermitian structures on a real hypersurface. J. Differential Geom. 13 (1978), 25–41. MR520599 Zbl 0379.53016CrossrefGoogle Scholar

About the article

Received: 2017-03-06

Revised: 2018-03-10

Published Online: 2019-07-04

Communicated by: P. Eberlein

Funding: This research was supported by Basic Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education (No. 2017R1A6A3A01012821) and the second author by grant Proj. No. NRF-2018-R1D1A1B-05040381 from the National Research Foundation of Korea.

Citation Information: Advances in Geometry, ISSN (Online) 1615-7168, ISSN (Print) 1615-715X, DOI: https://doi.org/10.1515/advgeom-2019-0012.

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