Glasnik Matematicki, Vol. 58, No. 1 (2023), 101-124. \( \)

SEMI-PARALLEL HOPF REAL HYPERSURFACES IN THE COMPLEX QUADRIC

Hyunjin Lee and Young Jin Suh

Department of Mathematics Education, Chosun University, Gwangju 61452, Republic of Korea
e-mail:lhjibis@hanmail.net

Department of Mathematics & RIRCM, Kyungpook National University, Daegu 41566, Republic of Korea
e-mail:yjsuh@knu.ac.kr

Abstract.   In this paper, we introduce the new notion of semi-parallel real hypersurface in the complex quadric \(Q^{m}\). Moreover, we give a nonexistence theorem for semi-parallel Hopf real hypersurfaces in the complex quadric \(Q^{m}\) for \(m \geq 3\).

2020 Mathematics Subject Classification.   53C40, 53C55

Key words and phrases.   Semi-parallel real hypersurface, semi-symmetric real hypersurface, singular normal vector field, complex structure, real structure, complex quadric

Full text (PDF) (access from subscribing institutions only)

https://doi.org/10.3336/gm.58.1.08

References:

  1. A. L. Besse, Einstein manifolds. Reprint of the 1987 edition, Springer-Verlag, Berlin, 2008.
    MathSciNet    CrossRef

  2. J. Berndt and Y. J. Suh, Real hypersurfaces in Hermitian symmetric spaces, De Gruyter, Berlin, 2022.
    MathSciNet    CrossRef

  3. D. E. Blair, Riemannian geometry of contact and symplectic manifolds. Second edition, Birkhäuser Boston, Ltd., Boston, 2010.
    MathSciNet    CrossRef

  4. J. Deprez, Semiparallel surfaces in Euclidean space, J. Geom. 25 (1985), 192–200.
    MathSciNet    CrossRef

  5. J. Deprez, Semiparallel hypersurfaces, Rend. Sem. Mat. Univ. Politec. Torino 44 (1986), 303–316.
    MathSciNet

  6. F. Dillen, Semi-parallel hypersurfaces of a real space form, Israel J. Math. 75 (1991), 193–202.
    MathSciNet    CrossRef

  7. S. Helgason, Differential geometry, Lie groups, and symmetric spaces. Corrected reprint of the 1978 original, American Mathematical Society, Providence, 2001.
    MathSciNet    CrossRef

  8. D. H. Hwang, H. Lee and C. Woo, Semi-parallel symmetric operators for Hopf hypersurfaces in complex two-plane Grassmannians, Monatsh. Math. 177 (2015), 539–550.
    MathSciNet    CrossRef

  9. S. Klein, Totally geodesic submanifolds of the complex quadric, Differential Geom. Appl. 26 (2008), 79–96.
    MathSciNet    CrossRef

  10. A. W. Knapp, Lie groups beyond an introduction. Second edition, Birkhäuser Boston, Inc., Boston, 2002.
    MathSciNet    CrossRef

  11. S. Kobayashi and K. Nomizu, Foundations of differential geometry. Vol. II. Reprint of the 1969 original, John Wiley & Sons, Inc., New York, 1996.
    MathSciNet

  12. H. Lee, J. D. Pérez and Y. J. Suh, Derivatives of normal Jacobi operator on real hypersurfaces in the complex quadric, Bull. Lond. Math. Soc. 52 (2020), 1122–1133.
    MathSciNet    CrossRef

  13. H. Lee and Y. J. Suh, Real hypersurfaces with recurrent normal Jacobi operator in the complex quadric, J. Geom. Phys. 123 (2018), 463–474.
    MathSciNet    CrossRef

  14. H. Lee and Y. J. Suh, Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric, Math. Phys. Anal. Geom. 23 (2020), Paper No. 44, 21 pp.
    MathSciNet    CrossRef

  15. H. Lee and Y. J. Suh, A new classification of real hypersurfaces with Reeb parallel structure Jacobi operator in the complex quadric, J. Korean Math. Soc. 58 (2021), 895–920.
    MathSciNet    CrossRef

  16. T.-H. Loo, Semi-parallel real hypersurfaces in complex two-plane Grassmannians, Differential Geom. Appl. 34 (2014), 87–102.
    MathSciNet    CrossRef

  17. C. J. G. Machado and J. D. Pérez, Real hypersurfaces in complex two-plane Grassmannians some of whose Jacobi operators are \(\xi\)-invariant, Internat. J. Math. 23 (2012), 1250002, 12 pp.
    MathSciNet    CrossRef

  18. R. Niebergall and P. J. Ryan, Semi-parallel and semi-symmetric real hypersurfaces in complex space forms, Kyungpook Math. J. 38 (1998), 227–234.
    MathSciNet

  19. M. Ortega, Classifications of real hypersurfaces in complex space forms by means of curvature conditions, Bull. Belg. Math. Soc. Simon Stevin 9 (2002), 351–360.
    MathSciNet    CrossRef

  20. J. D. Pérez, On the structure vector field of a real hypersurface in complex quadric, Open Math. 16 (2018), 185–189.
    MathSciNet    CrossRef

  21. J. D. Pérez, Commutativity of torsion and normal Jacobi operators on real hypersurfaces in the complex quadric, Publ. Math. Debrecen 95 (2019), 157–168.
    MathSciNet    CrossRef

  22. J. D. Pérez, Some real hypersurfaces in complex and complex hyperbolic quadrics, Bull. Malays. Math. Sci. Soc. 43 (2020), 1709–1718.
    MathSciNet    CrossRef

  23. J. D. Pérez and Y. J. Suh, Derivatives of the shape operator of real hypersurfaces in the complex quadric, Results Math. 73 (2018), Paper No. 126, 10 pp.
    MathSciNet    CrossRef

  24. J. D. Pérez and Y. J. Suh, Commutativity of Cho and normal Jacobi operators on real hypersurfaces in the complex quadric, Publ. Math. Debrecen 94 (2019), 359–367.
    MathSciNet    CrossRef

  25. H. Reckziegel, On the geometry of the complex quadric, in: Geometry and topology of submanifolds. VIII, World Sci. Publ., River Edge, 1996, 302–315.
    MathSciNet

  26. A. Romero, Some examples of indefinite complete complex Einstein hypersurfaces not locally symmetric, Proc. Amer. Math. Soc. 98 (1986), 283–286.
    MathSciNet    CrossRef

  27. A. Romero, On a certain class of complex Einstein hypersurfaces in indefinite complex space forms, Math. Z. 192 (1986), 627–635.
    MathSciNet    CrossRef

  28. B. Smyth, Differential geometry of complex hypersurfaces, Ann. of Math. (2) 85 (1967), 246–266.
    MathSciNet    CrossRef

  29. Z. I. Szabó, Structure theorems on Riemannian spaces satisfying \(R(X,Y)\cdot R=0\). I. The local version, J. Differential Geometry 17 (1982), 531–582.
    MathSciNet    CrossRef

  30. Z. I. Szabó, Structure theorems on Riemannian spaces satisfying \(R(X,Y)\cdot R=0\). II. Global versions, Geom. Dedicata 19 (1985), 65–108.
    MathSciNet    CrossRef

  31. Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape operator, Internat. J. Math. 25 (2014), 1450059, 17 pp.
    MathSciNet    CrossRef

  32. Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb invariant shape operator, Differential Geom. Appl. 38 (2015), 10–21.
    MathSciNet    CrossRef

  33. Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Adv. Math. 281 (2015), 886–905.
    MathSciNet    CrossRef

  34. Y. J. Suh, Real hypersurfaces in the complex quadric with harmonic curvature, J. Math. Pures Appl. (9) 106 (2016), 393–410.
    MathSciNet    CrossRef

  35. Y. J. Suh, Pseudo-anti commuting Ricci tensor and Ricci soliton real hypersurfaces in the complex quadric, J. Math. Pures Appl. (9) 107 (2017), 429–450.
    MathSciNet    CrossRef

  36. Y. Wang, Semi-symmetric almost coKähler 3-manifolds, Int. J. Geom. Methods Mod. Phys. 15 (2018), 1850031, 12 pp.
    MathSciNet    CrossRef

  37. Y. Wang, Nonexistence of Hopf hypersurfaces in complex two-plane Grassmannians with GTW parallel normal Jacobi operator, Rocky Mountain J. Math. 49 (2019), 2375–2393.
    MathSciNet    CrossRef

  38. Y. Wang, Some recurrent normal Jacobi operators on real hypersurfaces in complex two-plane Grassmannians, Publ. Math. Debrecen 95 (2019), 307–319.
    MathSciNet    CrossRef
Glasnik Matematicki Home Page