Kyungpook Mathematical Journal

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

DOI QR Code

Real Hypersurfaces with Invariant Normal Jacobi Operator in the Complex Hyperbolic Quadric

  • Jeong, Imsoon (Department of Mathematics Education, Cheongju University) ;
  • Kim, Gyu Jong (Department of Mathematics Education, Woosuk University)
  • Received : 2019.03.05
  • Accepted : 2019.06.10
  • Published : 2020.09.30
Download PDF
⟨ Previous Next ⟩

Abstract

We introduce the notion of Lie invariant normal Jacobi operators for real hypersurfaces in the complex hyperbolic quadric Qm∗ = SOom,2/SOmSO2. The invariant normal Jacobi operator implies that the unit normal vector field N becomes 𝕬-principal or 𝕬-isotropic. Then in each case, we give a complete classification of real hypersurfaces in Qm∗ = SOom,2/SOmSO2 with Lie invariant normal Jacobi operators.

Keywords

References

  1. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in complex quadrics, Internat. J. Math., 24(7)(2013), 1350050, 18 pp.
  2. A. L. Besse, Einstein manifolds, Springer-Verlag, Berlin, 2008.
  3. S. Helgason, Differential geometry, Lie groups and symmetric spaces, Graduate Studies in Mathematics 34, American Mathematical Society, 2001.
  4. I. Jeong, H. J. Kim and Y. J. Suh, Real hypersurfaces in complex two-plane Grassman-nians with parallel normal Jacobi operator, Publ. Math. Debrecen., 76(1-2)(2010), 203-218.
  5. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc., 296(1)(1986), 137-149. https://doi.org/10.1090/S0002-9947-1986-0837803-2
  6. M. Kimura, Some real hypersurfaces of a complex projective space, Saitama Math. J., 5(1987), 1-5.
  7. S. Klein, Totally geodesic submanifolds of the complex quadric, Differential Geom. Appl., 26(1)(2008), 79-96. https://doi.org/10.1016/j.difgeo.2007.11.004
  8. S. Klein, Totally geodesic submanifolds of the complex and the quaternionic 2-Grassmannians, Trans. Amer. Math. Soc., 361(9)(2009), 4927-4967. https://doi.org/10.1090/S0002-9947-09-04699-6
  9. S. Klein and Y. J. Suh, Contact real hypersurfaces in the complex hyperbolic quadric, Ann. Mat. Pura Appl., 198(4)(2019), 1481-1494 https://doi.org/10.1007/s10231-019-00827-y
  10. A. W. Knapp, Lie Groups beyond an introduction, Second edition, Progress in Mathematics 140, Birkhauser, 2002.
  11. S. Kobayashi and K. Nomizu, Foundations of differential geometry, Vol. II, A Wiley-Interscience Publication, Wiley Classics Library Ed., 1996.
  12. S. Montiel and A. Romero, Complex Einstein hypersurfaces of indefinite complex space forms, Math. Proc. Cambridge Philos. Soc., 94(3)(1983), 495-508. https://doi.org/10.1017/S0305004100000888
  13. J. D. Perez, Commutativity of Cho and structure Jacobi operators of a real hypersur-face in a complex projective space, Ann. Mat. Pura Appl., 194(6)(2015), 1781-1794. https://doi.org/10.1007/s10231-014-0444-0
  14. J. D. Perez and Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying ${\nabla}_U_iR=0$, Differential Geom. Appl., 7(3)(1997), 211-217. https://doi.org/10.1016/S0926-2245(97)00003-X
  15. J. D. Perez and Y. J. Suh, Certain conditions on the Ricci tensor of real hypersurfaces in quaternionic projective spaces, Acta Math. Hungar., 91(4)(2001), 343-356. https://doi.org/10.1023/A:1010676103922
  16. H. Reckziegel, On the geometry of the complex quadric, Geometry and Topology of Submanifolds VIII(Brussels/Nordfjordeid 1995), 302-315, World Sci. Publ., River Edge, NJ, 1996.
  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 of type B in complex two-plane Grassmannians, Monatsh. Math., 147(4)(2006), 337-355. https://doi.org/10.1007/s00605-005-0329-9
  22. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with parallel Riccitensor, Proc. Roy. Soc. Edinburgh Sect. A., 142(6)(2012), 1309-1324. https://doi.org/10.1017/S0308210510001472
  23. Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature, J. Math. Pures Appl., 100(1) (2013), 16-33. https://doi.org/10.1016/j.matpur.2012.10.010
  24. Y. J. Suh, Real hypersurfaces in the complex quadric with Reeb parallel shape operator, Internat. J. Math., 25(6)(2014), 1450059, 17 pp.
  25. 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
  26. Y. J. Suh, Real hypersurfaces in the complex quadric with harmonic curvature, J. Math. Pures Appl., 106(3) (2016), 393-410. https://doi.org/10.1016/j.matpur.2016.02.015
  27. Y. J. Suh, Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb Flow, Commun. Contemp. Math., 20(2)(2018), 1750031, 20 pp.
  28. Y. J. Suh, Real hypersurfaces in the complex hyperbolic quadric with parallel normal Jacobi operator, Mediterr. J. Math., 15(4)(2018), Paper No. 159, 14 pp. https://doi.org/10.1007/s00009-018-1202-0
  29. Y. J. Suh and D. H. Hwang, Real hypersurfaces in the complex quadric with commuting Ricci tensor, Sci. China Math., 59(11)(2016), 2185-2198. https://doi.org/10.1007/s11425-016-0067-7
  30. Y. J. Suh and D. H. Hwang, Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel shape operator, Ann. Mat. Pura Appl., 196(4)(2017), 1307--1326. https://doi.org/10.1007/s10231-016-0617-0
  31. 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(1-2)(2019), 75-107. https://doi.org/10.5486/PMD.2019.8262