Bulletin of the Korean Mathematical Society (대한수학회보)

Korean Mathematical Society (대한수학회)
권호 표지
DOI QR코드

DOI QR Code

A NEW CHARACTERIZATION OF TYPE (A) AND RULED REAL HYPERSURFACES IN NONFLAT COMPLEX SPACE FORMS

  • Wang, Yaning (School of Mathematics and Information Science Henan Normal University)
  • Received : 2021.07.12
  • Accepted : 2022.05.26
  • Published : 2022.07.31
Download PDF
⟨ Previous Next ⟩

Abstract

In this paper, we obtain an inequality involving the squared norm of the covariant differentiation of the shape operator for a real hypersurface in nonflat complex space forms. It is proved that the equality holds for non-Hopf case if and only if the hypersurface is ruled and the equality holds for Hopf case if and only if the hypersurface is of type (A).

Keywords

Acknowledgement

The author would like to thank the reviewer for his or her careful reading and comments.

References

  1. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132-141. https://doi.org/10.1515/crll.1989.395.132
  2. T. E. Cecil and P. J. Ryan, Geometry of hypersurfaces, Springer Monographs in Mathematics, Springer, New York, 2015. https://doi.org/10.1007/978-1-4939-3246-7
  3. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137-149. https://doi.org/10.2307/2000565
  4. M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z. 202 (1989), no. 3, 299-311. https://doi.org/10.1007/BF01159962
  5. S. H. Kon and T.-H. Loo, On characterizations of real hypersurfaces in a complex space form with η-parallel shape operator, Canad. Math. Bull. 55 (2012), no. 1, 114-126. https://doi.org/10.4153/CMB-2011-039-5
  6. S. Maeda, Some characterizations of the homogeneous ruled real hypersurface in a complex hyperbolic space, Mem. Grad. Sch. Sci. Eng. Shimane Univ. Ser. B Math. 52 (2019), 15-19.
  7. S. Maeda and H. Tanabe, A characterization of the homogeneous ruled real hypersurface in a complex hyperbolic space in terms of the first curvature of some integral curves, Arch. Math. (Basel) 105 (2015), no. 6, 593-599. https://doi.org/10.1007/s00013-015-0839-1
  8. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, in Tight and taut submanifolds (Berkeley, CA, 1994), 233-305, Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 1997.
  9. Y. J. Suh, On real hypersurfaces of a complex space form with η-parallel Ricci tensor, Tsukuba J. Math. 14 (1990), no. 1, 27-37. https://doi.org/10.21099/tkbjm/1496161316
  10. Y. J. Suh, Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995), no. 1, 63-79.
  11. Y. Wang, Cyclic η-parallel shape and Ricci operators on real hypersurfaces in two-dimensional nonflat complex space forms, Pacific J. Math. 302 (2019), no. 1, 335-352. https://doi.org/10.2140/pjm.2019.302.335
  12. Y. Wang, Remarks on η-parallel real hypersurfaces in ℂP2 and ℂH2, Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 5, 2050073, 15 pp. https://doi.org/10.1142/S0219887820500735