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Hopf Hypersurfaces in Complex Two-plane Grassmannians with Generalized Tanaka-Webster Reeb-parallel Structure Jacobi Operator

  • Kim, Byung Hak (Department of Applied Mathematics and Institute of Natural Sciences, Kyung Hee University) ;
  • Lee, Hyunjin (The Research Institute of Real and Complex Manifolds, Kyungpook National University) ;
  • Pak, Eunmi (Department of Mathematics, Kyungpook National University)
  • Received : 2017.12.28
  • Accepted : 2018.02.27
  • Published : 2019.09.23

Abstract

In relation to the generalized Tanaka-Webster connection, we consider a new notion of parallel structure Jacobi operator for real hypersurfaces in complex two-plane Grassmannians and prove the non-existence of real hypersurfaces in $G_2({\mathbb{C}}^{m+2})$ with generalized Tanaka-Webster parallel structure Jacobi operator.

Keywords

real hypersurface;complex two-plane Grassmannian;Hopf hypersurface;generalized Tanaka-Webster connection;structure Jacobi operator

Acknowledgement

Supported by : National Research Foundation of Korea(NRF)

References

  1. J. T. Cho, CR-structures on real hypersurfaces of a complex space form, Publ. Math. Debrecen, 54(1999), 473-487.
  2. J. T. Cho, Levi parallel hypersurfaces in a complex space form, Tsukuba J. Math., 30(2006), 329-343. https://doi.org/10.21099/tkbjm/1496165066
  3. I. Jeong, C. J. G. Machado, J. D. Perez, and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians with D-parallel structure Jacobi operator, Internat. J. Math., 22(5)(2011), 655-673. https://doi.org/10.1142/S0129167X11006957
  4. I. Jeong, J. D. Perez, and Y. J. Suh, Real hypersurfaces in complex two-plane Grass-mannians with parallel structure Jacobi operator, Acta Math. Hungar., 122(2009), 173-186. https://doi.org/10.1007/s10474-008-8004-y
  5. U-H. Ki, J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex space forms with $\xi$-parallel Ricci tensor and structure Jacobi operator, J. Korean Math. Soc., 44(2007), 307-326. https://doi.org/10.4134/JKMS.2007.44.2.307
  6. H. Lee, J. D. Perez, and Y. J. Suh, On the structure Jacobi operator of a real hyper-surface in complex projective space, Monatsh. Math., 158(2) (2009), 187-194. https://doi.org/10.1007/s00605-008-0025-7
  7. H. Lee, J. D. Perez, and Y. J. Suh, Real hypersurfaces in a complex projective space with pseudo-D-parallel structure Jacobi operator, Czechoslovak Math. J., 60(4)(2010), 1025-1036. https://doi.org/10.1007/s10587-010-0066-7
  8. H. Lee and Y. J. Suh, Real hypersurfaces of type B in complex two-plane Grassman-nians related to the Reeb vector, Bull. Korean Math. Soc., 47(3)(2010), 551-561. https://doi.org/10.4134/BKMS.2010.47.3.551
  9. C. J. G. Machado and J. D. Perez Real hypersurfaces in complex two-plane Grass-mannians some of whose Jacobi operators are $\xi$-invariant, Internat. J. Math., 23(3)(2012), 1250002, 12 pp.
  10. E. Pak and Y. J. Suh, Hopf hypersurfaces in complex two-plane Grassmannians with generalized Tanaka-Webster $D^{\perp}$-parallel structure Jacobi operator, Cent. Eur. J. Math., 12(2014), 1840-1851.
  11. D. V. Alekseevskii, Compact quaternion spaces, Funct. Anal. Appl., 2(1968), 106-114. https://doi.org/10.1007/BF01075944
  12. J. Berndt and Y. J. Suh, Real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math., 127(1999), 1-14. https://doi.org/10.1007/s006050050018
  13. J. Berndt and Y. J. Suh, Isometric flows on real hypersurfaces in complex two-plane Grassmannians, Monatsh. Math., 137(2002), 87-98. https://doi.org/10.1007/s00605-001-0494-4
  14. J. D. Perez and Y. J. Suh, Real hypersurfaces of quaternionic projective space satisfying ${\nabla}_U_iR=0$, Differential Geom. Appl., 7(1997), 211-217. https://doi.org/10.1016/S0926-2245(97)00003-X
  15. J. D. Perez and Y. J. Suh, Two conditions on the structure Jacobi operator for real hypersurfaces in complex projective space, Canad. Math. Bull., 54(3)(2011), 422-429. https://doi.org/10.4153/CMB-2011-020-4
  16. J. D. Perez, F. G. Santos, and Y. J. Suh, Real hypersurfaces in complex projective space whose structure Jacobi operator is D-parallel, Bull. Belg. Math. Soc. Simon Stevin, 13(2006), 459-469.
  17. N. Tanaka, On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Japan. J. Math., 20(1976), 131-190.
  18. S. Tanno, Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc., 314(1)(1989), 349-379. https://doi.org/10.1090/S0002-9947-1989-1000553-9
  19. S. M. Webster, Pseudo-Hermitian structures on a real hypersurface, J. Differential Geom., 13(1978), 25-41. https://doi.org/10.4310/jdg/1214434345