Journal of the Korean Mathematical Society (대한수학회지)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

GENERALIZED KILLING STRUCTURE JACOBI OPERATOR FOR REAL HYPERSURFACES IN COMPLEX HYPERBOLIC TWO-PLANE GRASSMANNIANS

  • Lee, Hyunjin (Research Institute of Real and Complex Manifold (RIRCM) Kyungpook National University) ;
  • Suh, Young Jin (Department of Mathematics & RIRCM Kyungpook National University) ;
  • Woo, Changhwa (Department of Applied Mathematics Pukyong National University)
  • Received : 2020.11.09
  • Accepted : 2021.06.18
  • Published : 2022.03.01
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, first we introduce a new notion of generalized Killing structure Jacobi operator for a real hypersurface M in complex hyperbolic two-plane Grassmannians SU2,m/S (U2·Um). Next we prove that there does not exist a Hopf real hypersurface in complex hyperbolic two-plane Grassmannians SU2,m/S (U2·Um) with generalized Killing structure Jacobi operator.

Keywords

Acknowledgement

This work was supported by grant Proj. No. NRF-2020-R1A2C1A-01101518 from National Research Foundation of Korea. The first author was supported by grant Proj. No. NRF-2019-R1I1A1A-01050300, the second author was supported by grant Proj. No. NRF-2018-R1D1A1B-05040381 and the third author was supported by the Pukyong National University Research Fund in 2019.

References

  1. J. F. Adams, Lectures on Exceptional Lie Groups, Chicago Lectures in Mathematics, University of Chicago Press, Chicago, IL, 1996.
  2. W. Ballmann, Lectures on Kahler manifolds, ESI Lectures in Mathematics and Physics, European Mathematical Society (EMS), Zurich, 2006. https://doi.org/10.4171/025
  3. J. Berndt and Y. J. Suh, Hypersurfaces in noncompact complex Grassmannians of rank two, Internat. J. Math. 23 (2012), no. 10, 1250103, 35 pp. https://doi.org/10.1142/S0129167X12501030
  4. J. Berndt and Y. J. Suh, Contact hypersurfaces in Kahler manifolds, Proc. Amer. Math. Soc. 143 (2015), no. 6, 2637-2649. https://doi.org/10.1090/S0002-9939-2015-12421-5
  5. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in Kahler manifolds, Commun. Contemp. Math. 23 (2021), no. 1, 1950039, 33 pp. https://doi.org/10.1142/s0219199719500391
  6. J. Berndt and Y. J. Suh, Real hypersurfaces with isometric Reeb flow in Kahler manifolds, Commun. Contemp. Math. 23 (2021), no. 1, 1950039, 33 pp. https://doi.org/10.1142/s0219199719500391
  7. J. Berndt and Y. J. Suh, Real hypersurfaces in hermitian symmetric spaces, Advances in Analysis and Geometry, Editor in Chief, Jie Xiao, Walter de Gruyter GmbH, Berlin/Boston (in press).
  8. D. E. Blair, Almost contact manifolds with Killing structure tensors, Pacific J. Math. 39 (1971), 285-292. http://projecteuclid.org/euclid.pjm/1102969563 https://doi.org/10.2140/pjm.1971.39.285
  9. A. Borel and J. De Siebenthal, Les sous-groupes ferm'es de rang maximum des groupes de Lie clos, Comment. Math. Helv. 23 (1949), 200-221. https://doi.org/10.1007/BF02565599
  10. K. Heil, A. Moroianu, and U. Semmelmann, Killing and conformal Killing tensors, J. Geom. Phys. 106 (2016), 383-400. https://doi.org/10.1016/j.geomphys.2016.04.014
  11. S. Helgason, Differential geometry, Lie groups, and symmetric spaces, corrected reprint of the 1978 original, Graduate Studies in Mathematics, 34, American Mathematical Society, Providence, RI, 2001. https://doi.org/10.1090/gsm/034
  12. I. Jeong, C. J. G. Machado, J. D. P'erez, and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with 𝕯⊥-parallel structure Jacobi operator, Internat. J. Math. 22 (2011), no. 5, 655-673. https://doi.org/10.1142/S0129167X11006957
  13. H. Lee, S. Kim, and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with certain commuting condition II, Czechoslovak Math. J. 64(139) (2014), no. 1, 133-148. https://doi.org/10.1007/s10587-014-0089-6
  14. R.-H. Lee and T.-H. Loo, Hopf hypersurfaces in complex Grassmannians of rank two, Results Math. 71 (2017), no. 3-4, 1083-1107. https://doi.org/10.1007/s00025-016-0601-4
  15. 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.
  16. 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), no. 5, 3389-3407. https://doi.org/10.1007/s00009-016-0692-x
  17. H. Lee, Y. J. Suh, and C. Woo, Quadratic Killing structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians, arXiv:2010.13267 [math.DG].
  18. H. Lee, C. Woo, and Y. J. Suh, Quadratic Killing normal Jacobi operator for real hypersurfaces in complex Grassmannians of rank 2, J. Geom. Phys. 160 (2021), 103975, 14 pp. https://doi.org/10.1016/j.geomphys.2020.103975
  19. A. Martinez and J. D. Perez, Real hypersurfaces in quaternionic projective space, Ann. Mat. Pura Appl. (4) 145 (1986), 355-384. https://doi.org/10.1007/BF01790548
  20. J. D. Perez, Cyclic-parallel real hypersurfaces of quaternionic projective space, Tsukuba J. Math. 17 (1993), no. 1, 189-191. https://doi.org/10.21099/tkbjm/1496162139
  21. J. D. Perez, Comparing Lie derivatives on real hypersurfaces in complex projective spaces, Mediterr. J. Math. 13 (2016), no. 4, 2161-2169. https://doi.org/10.1007/s00009-015-0601-8
  22. J. D. Perez and Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying ∇UiR = 0, Differential Geom. Appl. 7 (1997), no. 3, 211-217. https://doi.org/10.1016/S0926-2245(97)00003-X
  23. J. D. Perez and Y. J. Suh, The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians, J. Korean Math. Soc. 44 (2007), no. 1, 211-235. https://doi.org/10.4134/JKMS.2007.44.1.211
  24. J. D. Perez, Y. J. Suh, and Y. Watanabe, Generalized Einstein real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys. 60 (2010), no. 11, 1806-1818. https://doi.org/10.1016/j.geomphys.2010.06.017
  25. U. Semmelmann, Conformal Killing forms on Riemannian manifolds, Math. Z. 245 (2003), no. 3, 503-527. https://doi.org/10.1007/s00209-003-0549-4
  26. Y. J. Suh, Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians, Adv. in Appl. Math. 50 (2013), no. 4, 645-659. https://doi.org/10.1016/j.aam.2013.01.001
  27. Y. J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with Reeb vector field, Adv. in Appl. Math. 55 (2014), 131-145. https://doi.org/10.1016/j.aam.2014.01.005
  28. Y. J. Suh, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with commuting Ricci tensor, Internat. J. Math. 26 (2015), no. 1, 1550008, 26 pp. https://doi.org/10.1142/S0129167X15500081
  29. Y. J. Suh, Generalized Killing Ricci tensor for real hypersurfaces in complex two-plane Grassmannians, J. Geom. Phys. 159 (2021), 103799, 15 pp. https://doi.org/10.1016/j.geomphys.2020.103799
  30. Y. J. Suh, Generalized Killing-Ricci tensor for real hypersurfaces in complex hyperbolic two-plane Grassmannians, Mediterr. J. Math. 18 (2021), no. 3, Paper No. 88, 28 pp. https://doi.org/10.1007/s00009-021-01724-6
  31. Y. J. Suh and C. Woo, Real hypersurfaces in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor, Math. Nachr. 287 (2014), no. 13, 1524-1529. https://doi.org/10.1002/mana.201300283
  32. G. Thompson, Killing tensors in spaces of constant curvature, J. Math. Phys. 27 (1986), no. 11, 2693-2699. https://doi.org/10.1063/1.527288
  33. K. Yano, On harmonic and Killing vector fields, Ann. of Math. (2) 55 (1952), 38-45. https://doi.org/10.2307/1969418
  34. K. Yano, Harmonic and Killing tensor fields in Riemannian spaces with boundary, J. Math. Soc. Japan 10 (1958), 430-437. https://doi.org/10.2969/jmsj/01040430
  35. K. Yano, Harmonic and Killing vector fields in compact orientable Riemannian spaces with boundary, Ann. of Math. (2) 69 (1959), 588-597. https://doi.org/10.2307/1970024