Research ArticleNo Access

Remarks on $η$ -parallel real hypersurfaces in $ℂ P^{2}$ and $ℂ H^{2}$

School of Mathematics and Information Science, Henan Normal University, Xinxiang 453007, Henan, P. R. China

https://doi.org/10.1142/S0219887820500735Cited by:0 (Source: Crossref)

Abstract

Let $M$ be a three-dimensional real hypersurface in a nonflat complex space form of complex dimension two. In this paper, we prove that $M$ is $η$ -parallel with two distinct principal curvatures at each point if and only if it is locally congruent to a geodesic sphere in $ℂ P^{2}$ or a horosphere, a geodesic sphere or a tube over totally geodesic complex hyperbolic plane in $ℂ H^{2}$ . Moreover, $η$ -parallel real hypersurfaces in $ℂ P^{2}$ and $ℂ H^{2}$ under some other conditions are classified and these results extend Suh’s in [Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79] and Kon–Loo’s in [On characterizations of real hypersurfaces in a complex space form with $η$ -parallel shape operator, Canad. Math. Bull. 55 (2012) 114–126].

Keywords:

AMSC: 53B25, 53D15

References

1. S. S. Ahn, S. B. Lee and Y. J. Suh, On ruled real hypersurfaces in a complex space form, Tsukuba J. Math. 17 (1993) 311–322. Crossref, Google Scholar
2. J. Berndt, Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989) 132–141. Web of Science, Google Scholar
3. J. Berndt and J. C. Diaz-Ramos, Real hypersurfaces with constant principal curvatures in the complex hyperbolic plane, Proc. Amer. Math. Soc. 135 (2007) 3349–3357. Crossref, Web of Science, Google Scholar
4. T. E. Cecil and P. J. Ryan, Geometry of Hypersurfaces (Springer, New York, 2015). Crossref, Google Scholar
5. J. C. Díaz-Ramos, M. Domínguez-Vázquez and C. Vidal-Castin̋eira, Real hypersurfaces with two principal curvatures in complex projective and hyperbolic planes, J. Geom. Anal. 27 (2017) 442–465. Crossref, Web of Science, Google Scholar
6. J. C. Díaz-Ramos and C. Vidal-Castin̋eira, Examples of real hypersurfaces with two principal curvatures in complex projective planes, Int. J. Geom. Methods Mod. Phys. 12 (2015) 1560013. Link, Web of Science, Google Scholar
7. T. A. Ivey and P. J. Ryan, Real hypersurfaces in $ℂ P^{2}$ and $ℂ H^{2}$ with two distrinct principal curvatures, Glasgow Math. J. 58 (2016) 137–152. Crossref, Web of Science, Google Scholar
8. M. Kimura, Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986) 137–149. Crossref, Web of Science, Google Scholar
9. M. Kimura, Sectional curvatures of holomorphic planes on a real hypersurfaces in $P^{n} (C)$ , Math. Ann. 276 (1987) 487–497. Crossref, Web of Science, Google Scholar
10. M. Kimura and S. Maeda, On real hypersurfaces of a complex projective space, Math. Z. 202 (1989) 299–311. Crossref, Web of Science, Google Scholar
11. M. Kon, Pseudo-Einstein real hypersurfaces in complex space forms, J. Differ. Geom. 14 (1979) 339–354. Crossref, Google Scholar
12. 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) 114–126. Crossref, Web of Science, Google Scholar
13. S. H. Kon and T. H. Loo, Real hypersurfaces in a complex space form with $η$ -parallel shape operator, Math. Z. 269 (2011) 47–58. Crossref, Web of Science, Google Scholar
14. M. Lohnherr and H. Reckziegel, On ruled real hypersurfaces in complex space forms, Geom. Dedicata 74 (1999) 267–286. Crossref, Web of Science, Google Scholar
15. S. Montiel and A. Romero, On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (1986) 245–261. Crossref, Web of Science, Google Scholar
16. R. Niebergall and P. J. Ryan, Real hypersurfaces in complex space forms, Tight and Taut Submanifolds, Mathematical Sciences Research Institute Publications, Vol. 32 (Cambridge University Press, Cambridge, 1997), pp. 233–305. Google Scholar
17. K. Okumura, Non-homogeneous $η$ -Einstein real hypersurfaces in a $2$ -dimensional nonflat complex space form, in Proc. Workshop on Differential Geometry of Submanifolds and Its Related Topics (Saga, 2012), pp. 98–112. Google Scholar
18. M. Okumura, Contact hypersurfaces in certain Kaehlerian manifolds, Tohoku Math. J. 18 (1966) 74–102. Crossref, Google Scholar
19. M. Okumura, On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212 (1975) 355–364. Crossref, Web of Science, Google Scholar
20. K. Panagiotidou and P. J. Xenos, Real hypersurfaces in $ℂ P^{2}$ and $ℂ H^{2}$ whose structure Jacobi operator is Lie $𝔻$ -parallel, Note Mat. 32 (2012) 89–99. Google Scholar
21. Y. J. Suh, On real hypersurfaces of a complex space form with $η$ -parallel Ricci tensor, Tsukuba J. Math. 14 (1990) 27–37. Crossref, Google Scholar
22. Y. J. Suh, Characterizations of real hypersurfaces in complex space forms in terms of Weingarten map, Nihonkai Math. J. 6 (1995) 63–79. Google Scholar
23. R. Takagi, On homogeneous real hypersurfaces in a complex projective space, Osaka J. Math. 10 (1973) 495–506. Google Scholar
24. R. Takagi, Real hypersurfaces in a complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975) 43–53. Crossref, Web of Science, Google Scholar
25. Y. Wang, Cyclic $η$ -parallel shape and Ricci operators on real hypersurfaces in two-dimensional nonflat complex space forms, Pacific J. Math. 302 (2019) 335–352. Crossref, Web of Science, Google Scholar