# NOTES ON REAL HYPERSURFACES IN A COMPLEX SPACE FORM

• Cho, Jong Taek
• Published : 2015.01.31
• 31 8

#### Abstract

We characterize a homogeneous real hypersurface of type (A) or a ruled real hypersurface in a non-flat complex space form, respectively.

#### Keywords

real hypersurface;complex space form;almost contact structure

#### References

1. T. Adachi, M. Kameda, and S. Maeda, Real hypersurfaces which are contact in a nonflat complex space form, Hokkaido Math. J. 40 (2011), no. 2, 205-217. https://doi.org/10.14492/hokmj/1310042828
2. S.-S. Ahn, S.-B. Lee, and Y. J. Suh, On ruled real hypersurfaces in a complex space form, Tsukuba J. Math. 17 (1993), no. 2, 311-322. https://doi.org/10.21099/tkbjm/1496162264
3. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132-141.
4. D. E. Blair, Riemannian Geometry of Contact and Symplectic Manifolds, Progress in Math. 203, Birkhauser Boston, Inc., Boston, second edition, 2010.
5. D. E. Blair, Almost contact manifolds with Killing structure tensors, Pacific J. Math. 39 (1971), no. 2, 285-292. https://doi.org/10.2140/pjm.1971.39.285
6. T. E. Cecil and P. J. Ryan, Focal sets and real hypersurfaces in complex projective space, Trans. Amer. Math. Soc. 269 (1982), no. 2, 481-499.
7. J. T. Cho and J. Inoguchi, Contact metric hypersurfaces in complex space forms, Proceedings of the workshop on Differential Geometry of Submanifolds and Related Topics, Saga, August 4-6, 2012.
8. J. T. Cho and M. Kimura, Transversal symmetries on real hypersurfaces in a complex space form, Hiroshima Math. J. 43 (2013), no. 2, 223-238.
9. G. Dileo and A. M. Pastore, Almost Kenmotsu manifolds and local symmetry, Bull. Belg. Math. Soc. Simon Stevin 14 (2007), no. 2, 343-354.
10. S. I. Goldberg and K. Yano, Integrability of almost cosymplectic structures, Pacific J. Math. 31 (1969), 373-381. https://doi.org/10.2140/pjm.1969.31.373
11. S. Kanemaki, Quasi-Sasakian manifolds, Tohoku Math. J. 29 (1977), no. 2, 227-233. https://doi.org/10.2748/tmj/1178240654
12. K. Kenmotsu, A class of almost contact Riemannian manifolds, Tohoku Math. J. 24 (1972), 93-103. https://doi.org/10.2748/tmj/1178241594
13. U.-H. Ki and Y. J. Suh, On real hypersurfaces of a complex space form, Math. J. Okayama Univ. 32 (1990), 207-221.
14. U.-H. Ki and Y. J. Suh, On a characterization of real hypersurfaces of type A in a complex space form, Canad. Math. Bull. 37 (1994), no. 2, 238-244. https://doi.org/10.4153/CMB-1994-035-8
15. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), no. 1, 137-149. https://doi.org/10.1090/S0002-9947-1986-0837803-2
16. M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurfaces in $P^n({\mathbb{C}})$, Math. Ann. 276 (1987), no. 3, 487-497. https://doi.org/10.1007/BF01450843
17. M. Kon, Pseudo-Einstein real hypersurfaces in complex space forms, J. Diffential Geometry 14 (1979), no. 3, 339-354. https://doi.org/10.4310/jdg/1214435100
18. S. H. Kon and T. H. Loo, Real hypersurfaces in a complex space form with ${\eta}$-parallel shape operator, Math. Z. 269 (2011), no. 1-2, 47-58. https://doi.org/10.1007/s00209-010-0715-4
19. Y. Maeda, On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), no. 3, 529-540. https://doi.org/10.2969/jmsj/02830529
20. S. Maeda and S. Udagawa, Real hypersurfaces of a complex projective space in terms of holomorphic distribution, Tsukuba J. Math. 14 (1990), no. 1, 39-52. https://doi.org/10.21099/tkbjm/1496161317
21. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (1986), no. 2, 245-261. https://doi.org/10.1007/BF00164402
22. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and taut submanifolds (Berkeley, CA, 1994), 233-305, Math. Sci. Res. Inst. Publ., 32, Cambridge Univ. Press, Cambridge, 1997.
23. M. Okumura, Certain almost contact hypersurfaces in Kaehlerian manifolds of constant holomorphic sectional curvature, Tohoku Math. J. (2) 16 (1964), 270-284. https://doi.org/10.2748/tmj/1178243673
24. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975), 355-364. https://doi.org/10.1090/S0002-9947-1975-0377787-X
25. M. Okumura, Compact real hypersurfaces of a complex projective space, J. Differential Geom. 12 (1977), no. 4, 595-598. https://doi.org/10.4310/jdg/1214434228
26. Z. Olszak, Curvature properties of quasi-Sasakian manifolds, Tensor (N.S.) 38 (1982), 19-28.
27. Y. J. Suh, On real hypersurfaces of a complex space form with ${\eta}$-parallel Ricci tensor, Tsukuba J. Math. 14 (1990), no. 1, 27-37. https://doi.org/10.21099/tkbjm/1496161316
28. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 19 (1973), 495-506.
29. R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures I, II, J. Math. Soc. Japan 15 (1975), 43-53, 507-516.

#### Cited by

1. CONFORMALLY FLAT NORMAL ALMOST CONTACT 3-MANIFOLDS vol.38, pp.1, 2016, https://doi.org/10.5831/HMJ.2016.38.1.59
2. Real hypersurfaces with Killing type operators in a nonflat complex space form 2017, https://doi.org/10.1007/s00022-017-0375-1
3. Certain types of real hypersurfaces in complex space forms vol.109, pp.1, 2018, https://doi.org/10.1007/s00022-018-0405-7

#### Acknowledgement

Supported by : Chonnam National University