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ON THE NON-EXISTENCE OF REAL HYPERSURFACES IN COMPLEX HYPERBOLIC TWO-PLANE GRASSMANNIANS WITH GTW-RECURRENT STRUCTURE JACOBI OPERATOR

  • WOE HYUN KIM (Department of Applied Mathematics, Pukyong National University)
  • Received : 2026.03.16
  • Accepted : 2026.05.23
  • Published : 2026.05.30

Abstract

In this paper, we have introduced a new notion of GTW recurrent structure Jacobi of real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S (U2 · Um). Next, we show a non-existence property of real hypersurfaces in complex hyperbolic two-plane Grassmannians SU2,m/S (U2 · Um) satisfying such a curvature condition.

Keywords

References

  1. D.E. Blair, Almost contact manifolds with Killing structure tensors, Pacific J. Math. 39 (1971), 285-292. https://doi.org/10.2140/pjm
  2. J. Berndt and Y.J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math. 23 (2012), 35 pages. https://doi.org/10.1142/S0129167X12501030
  3. J. Berndt and Y.J. Suh, Real Hypersurfaces in Hermitian Symmetric Spaces, Advances in Analysis and Geometry, Walter de Gruyter GmbH, Berlin, Boston, 2021.
  4. J.T. Cho, CR structures on real hypersurfaces of a complex space form, Publ. Math. Debrecen 54 (1999), 473-487. https://doi.org/10.5486/PMD
  5. J.T. Cho, Levi parallel hypersurfaces in a complex space form, Tsukuba J. Math. 30 (2006), 329-344. https://doi.org/10.21099/tkbjm/1496165066
  6. T.E. Cecil and P.J. Ryan, Geometry of Hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015.
  7. A. De and T.H. Loo, Pseudo parallel CR-submanifolds in a non-flat complex space form, Taiwanese J. Math. 18 (2014), 1549-1562. https://doi.org/10.11650/tjm.18.2014.4034
  8. P.B. Eberlein, Geometry of Nonpositively Curved Manifolds, University of Chicago Press, Chicago, 1996.
  9. D.H. Hwang and C. Woo, Semisymmetric hypersurfaces in complex hyperbolic two-plane Grassmannians, Monatsh. Math. 176 (2015), 2377-2393.
  10. I. Jeong and C. Woo, Recurrent structure Jacobi operator of real hypersurfaces in complex hyperbolic two-plane Grassmannians, J. Appl. Math. Inform. 39 (2021), 327-338.
  11. R.-H. Lee and T.-H. Loo, Hopf hypersurfaces in complex Grassmannians of rank two, Results Math. 71 (2017), 1083-1107. https://doi.org/10.1007/s00025-016-0601-4
  12. H. Lee, Y.J. Suh and C. Woo, Reeb parallel Ricci tensor for homogeneous real hypersurfaces in complex hyperbolic two-plane Grassmannians, Math. Nachr. 288 (2015), 1-12.
  13. H. Lee, Y.J. Suh and C. Woo, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting structure Jacobi operators, Mediterr. J. Math. 13 (2016), 3389-3407. https://doi.org/10.1007/s00009-016-0692-x
  14. J.D. Pérez and F.G. Santos, Real hypersurfaces in complex projective space whose structure Jacobi operator is cyclic-Ryan parallel, Kyungpook Math. J. 49 (2009), 211-219. https://doi.org/10.5666/KMJ.2009.49.2.211
  15. J. D. Perez, Y. J. Suh and C. Woo, Real hypersurfaces in complex hyperbolic two-plane Grassmann ions with commating shape operator, Open Math. 13 (2015), 493-501.
  16. J.D. Perez and F.G. Santos, Real hypersurfaces in complex projective space with recurrent structure Jacobi operator, Differential Geometry and its Applications 26 (2008), 218-223. https://doi.org/10.1016/j.difgeo.2007.11.015
  17. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, Vol.1, Wiley, New York, 1963, 304-305.
  18. Y. J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Adv. Appl. Math. 50 (2013), 645-659. https://doi.org/10.1016/j.aam.2013.01.001
  19. Y. J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmann ions with Reeb vectorfeld, Adv. Appl. Math. 55 (2014), 131-145. https://doi.org/10.1016/j.aam.2014.01.005
  20. Y. J. Suh and C. Woo, Real hypersurfaces in complex hyperbolic two-plane Grassmann ions with parallel Ricci tensor, Math. Nachr. 287 (2014), 1524-1529. https://doi.org/10.1002/mana.v287.13
  21. N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math. 20 (1976), 131-190. https://doi.org/10.4099/math1924.2.131
  22. S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349-379. https://doi.org/10.1090/tran/1989-314-01
  23. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 19 (1973), 495-506.
  24. S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25-41. https://doi.org/10.4310/jdg/1214434345