Kyungpook Mathematical Journal

Department of Mathematics, Kyungpook National University (경북대학교 자연과학대학 수학과)
권호 표지
DOI QR코드

DOI QR Code

Cyclic Structure Jacobi Semi-symmetric Real Hypersurfaces in the Complex Hyperbolic Quadric

  • Imsoon Jeong (Department of Mathematics Education, Cheongju University) ;
  • Young Jin Suh (Department of Mathmatics & RIRCM, Kyungpook National University)
  • Received : 2022.10.24
  • Accepted : 2023.03.29
  • Published : 2023.06.30
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we introduce the notion of cyclic structure Jacobi semi-symmetric real hypersurfaces in the complex hyperbolic quadric Qm* = SO02,m/SO2SOm. We give a classifiction of when real hypersurfaces in the complex hyperbolic quadric Qm* having 𝔄-principal or 𝔄-isotropic unit normal vector fields have cyclic structure Jacobi semi-symmetric tensor.

Keywords

Acknowledgement

The first author was supported by grant Proj. No. NRF-2021-R1F1A-1064192, and the second by NRF-2018-R1D1A1B-05040381 and NRF-2021-R1C1C-2009847 from National Research Foundation of Korea.

References

  1. J. Berndt and Y. J. Suh, Real Hypersurfaces in Hermitian Symmetric Spaces, Adv. Anal. Geom., De Gruyter , Berlin(2022).
  2. A. L. Besse, Einstein Manifolds, Springer-Verlag(2008).
  3. S. K. Chaubey, Y. J. Suh and U. C. De, Characterizations of the Lorentzian manifolds admitting a type of semi-symmetric connection, Anal. Math. Phys., 10(2020), 15 pp.
  4. I. Jeong and Y. J. Suh, Real hypersurfaces in the complex quadric with Killing structure Jacobi operator, J. Geom. Phys., 139(2019), 88-102. https://doi.org/10.1016/j.geomphys.2018.09.021
  5. I. Jeong, E. Pak and Y. J. Suh, Real hypersurfaces in the complex hyperbolic quadric with normal Jacobi operator of Codazzi type, Internat. J. Math., 32(14)(2021), 23 pp.
  6. S. Klein and Y. J. Suh, Contact real hypersurfaces in the complex hyperbolic quadric, Anal. Mate. Pura Appl., 198(2019), 1481-1494. https://doi.org/10.1007/s10231-019-00827-y
  7. A. W. Knapp, Lie Groups Beyond an Introduction, Progress in Math., Birkhauser(2002).
  8. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, A Wiley-Interscience Publ., Wiley Classics Library Ed.(1996).
  9. H. Lee and Y. J. Suh, Commuting Jacobi operators on real hypersurfaces of Type B in the complex quadric, Math. Phys. Anal. Geom., 23(4)(2020), 21 pp.
  10. S. Montiel and A. Romero, Complex Einstein hypersurfaces of indefinite complex space forms, Math. Proc. Cambridge Philos. Soc., 94(1983), 495-508. https://doi.org/10.1017/S0305004100000888
  11. K. Nomizu, Invariant affine connections on homogeneous spaces, Amer. J. Math., 76(1954), 33-65. https://doi.org/10.2307/2372398
  12. J. D. Perez, Cyclic-parallel real hypersurfaces of quaternionic projective space, Tsukuba J. Math., 17(1)(1993), 189-191. https://doi.org/10.21099/tkbjm/1496162139
  13. J. D. Perez and F. G. Santos, Real Hypersurfaces in complex projective space whose structure Jacobi operator is cyclic-Ryan parallel, Kyungpook Math. J., 49(2)(2009), 211-219. https://doi.org/10.5666/KMJ.2009.49.2.211
  14. J. D. Perez and Y. J. Suh, Derivatives of the shape operator of real hypersurfaces in the complex quadric, Results Math., 73(3)(2018), 10 pp.
  15. J. D. Perez, F. G. Santos and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is Lie ξ-parallel, Differential Geom. Appl., 22(2)(2005), 181-188. https://doi.org/10.1016/j.difgeo.2004.10.005
  16. H. Reckziegel, On the geometry of the complex quadric, World Sci. Publ., River Edge, NJ, (1995).
  17. A. Romero, Some examples of indefinite complete complex Einstein hypersurfaces not locally symmetric, Proc. Amer. Math. Soc., 98(2)(1986), 283-286. https://doi.org/10.1090/S0002-9939-1986-0854034-6
  18. A. Romero, On a certain class of complex Einstein hypersurfaces in indefinite complex space forms, Math. Z., 192(4)(1986), 627-635. https://doi.org/10.1007/BF01162709
  19. B. Smyth, Differential geometry of complex hypersurfaces, Ann. of Math., 85(1967), 246-266. https://doi.org/10.2307/1970441
  20. B. Smyth, Homogeneous complex hypersurfaces, J. Math. Soc. Japan, 20(1968), 643-647. https://doi.org/10.2969/jmsj/02040643
  21. Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape operator, Internat. J. Math. 25(6)(2014), 17 pp.
  22. Y. J. Suh, Real hypersurfaces in the complex quadric with parallel Ricci tensor, Adv. Math., 281(2015), 886-905. https://doi.org/10.1016/j.aim.2015.05.012
  23. Y. J. Suh, Real hypersurfaces in the complex quadric with parallel structure Jacobi operator, Differential. Geom. Appl., 51(2017), 33-48. https://doi.org/10.1016/j.difgeo.2017.01.001
  24. Y. J. Suh, Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow, Commun. Contemp. Math., 20(2)(2018), 20 pp.
  25. Y. J. Suh, Real hypersurfaces in the complex quadric whose structure Jacobi operator is of Codazzi type, Math. Nachr., 292(6)(2019), 1375-1391. https://doi.org/10.1002/mana.201800184
  26. Y. J. Suh, Cyclic structure Jacobi semi-symmetric real hypersurfaces in the complex quadric, Math. Z., 303(2023), 20pp.
  27. Y. J. Suh, J. D. Perez and C. Woo, Real hypersurfaces in the complex hyperbolic quadric with parallel structure Jacobi operator, Publ. Math. Debrecen, 94(2019), 75-107. https://doi.org/10.5486/PMD.2019.8262
  28. Z. I. Szabo, Structure theorems on Riemannian spaces satisfying R(X, Y)R = 0, I, The local version, J. Differential Geom., 17(4)(1982), 531-582. https://doi.org/10.4310/jdg/1214437486