Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
-
Yaning Wang
Abstract
In this paper, Hopf hypersurfaces in a complex projective plane ℂP2(c) or a complex hyperbolic plane ℂH2(c) with constant scalar curvature are classified. For a non-Hopf hypersurface in ℂP2(c) with constant scalar curvature r, it is proved that if the structure vector field is an eigenvector of the Ricci operator, then either r = 7c/2 or r = 3c/2. Moreover, these two cases are determined completely under an additional condition.
Acknowledgements
The author would like to express his sincere thanks for many valuable suggestions in the reviewer’s report.
-
Funding: This work has been supported by the Key Scientific Research Program in Universities of Henan Province (No. 20A110023) and the Fostering Foundation of National Foundation in Henan Normal University (No. 2019PL22).
Communicated by: P. Eberlein
References
[1] T. E. Cecil, P. J. Ryan, Geometry of hypersurfaces. Springer 2015. MR3408101 Zbl 1331.5300110.1007/978-1-4939-3246-7Search in Google Scholar
[2] J. T. Cho, M. Kimura, Real hypersurfaces with constant φ-sectional curvature in complex projective space. Differential Geom. Appl. 68 (2020), 101573, 11. MR4016112 Zbl 1435.5304410.1016/j.difgeo.2019.101573Search in Google Scholar
[3] J. C. Díaz-Ramos, M. Domínguez-Vázquez, C. Vidal-Castiñeira, Strongly 2-Hopf hypersurfaces in complex projective and hyperbolic planes. Ann. Mat. Pura Appl. (4) 197 (2018), 469–486. MR3772912 Zbl 1387.5302910.1007/s10231-017-0687-7Search in Google Scholar
[4] T. A. Ivey, P. J. Ryan, Hopf hypersurfaces of small Hopf principal curvature in ℂH2. Geom. Dedicata 141 (2009), 147–161. MR2520069 Zbl 1177.5304510.1007/s10711-008-9349-7Search in Google Scholar
[5] T. A. Ivey, P. J. Ryan, Hypersurfaces in ℂP2 and ℂH2 with two distinct principal curvatures. Glasg. Math. J. 58 (2016), 137–152. MR3426432 Zbl 1337.5306710.1017/S0017089515000105Search in Google Scholar
[6] H. S. Kim, P. J. Ryan, A classification of pseudo-Einstein hypersurfaces in ℂP2. Differential Geom. Appl. 26 (2008), 106–112. MR2393977 Zbl 1143.5305010.1016/j.difgeo.2007.11.007Search in Google Scholar
[7] M. Kon, 3-dimensional real hypersurfaces and Ricci operator. Differ. Geom. Dyn. Syst. 16 (2014), 189–202. MR3226614 Zbl 1331.53030Search in Google Scholar
[8] M. Kon, Ricci tensor of real hypersurfaces. Pacific J. Math. 281 (2016), 103–123. MR3459968 Zbl 1333.5308010.2140/pjm.2016.281.103Search in Google Scholar
[9] M. Lohnherr, On ruled real hypersurfaces of complex space forms. PhD thesis, University of Cologne, 1988.Search in Google Scholar
[10] M. Lohnherr, H. Reckziegel, On ruled real hypersurfaces in complex space forms. Geom. Dedicata 74 (1999), 267–286. MR1669351 Zbl 0932.5301810.1023/A:1005000122427Search in Google Scholar
[11] R. Niebergall, P. J. Ryan, Real hypersurfaces in complex space forms. In: Tight and taut submanifolds (Berkeley, CA, 1994), volume 32 of Math. Sci. Res. Inst. Publ., 233–305, Cambridge Univ. Press 1997. MR1486875 Zbl 0904.53005Search in Google Scholar
[12] K. Okumura, Non-homogeneous η-Einstein real hypersurfaces in a 2-dimensional nonflat complex space form. In: Differential geometry of submanifolds and its related topics, 98–112, World Sci. Publ., Hackensack, NJ 2014. MR3203475 Zbl 1302.5306710.1142/9789814566285_0009Search in Google Scholar
[13] K. Panagiotidou, P. J. Xenos, Real hypersurfaces in ℂP2 and ℂH2 whose structure Jacobi operator is Lie 𝔻-parallel. Note Mat. 32 (2012), 89–99. MR3071797 Zbl 1282.53049Search in Google Scholar
[14] P. Petersen, Riemannian geometry. Springer 2006. MR2243772 Zbl 1220.53002Search in Google Scholar
[15] T. Sasahara, Ricci curvature of real hypersurfaces in non-flat complex space forms. Mediterr. J. Math. 15 (2018), Paper No. 141, 12 pages. MR3808568 Zbl 1394.5302710.1007/s00009-018-1183-zSearch in Google Scholar
[16] 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. MR4028776 Zbl 1434.5302110.2140/pjm.2019.302.335Search in Google Scholar
[17] Y. Wang, Generalized 𝓓-Einstein real hypersurfaces in ℂP2 and ℂH2. Canad. Math. Bull. 63 (2020), 909–920. MR4176779 Zbl 1457.5301110.4153/S0008439520000156Search in Google Scholar
© 2022 Walter de Gruyter GmbH, Berlin/Boston
Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
Articles in the same Issue
- Frontmatter
- On weak Fano manifolds with small contractions obtained by blow-ups of a product of projective spaces
-
A geographical study of
- A characterization of centrally symmetric convex bodies in terms of visual cones
- Closed Majorana representations of {3, 4}+-transposition groups
- Real hypersurfaces in ℂP2 and ℂH2 with constant scalar curvature
- Positive semigroups in lattices and totally real number fields
- A new axiomatics for masures II
- Helmut Salzmann and his legacy
- Some remarks on proper actions, proper metric spaces, and buildings
- Regular parallelisms on PG(3,ℝ) from generalized line stars: the oriented case
- A note on the Kleinewillinghöfer types of 4-dimensional Laguerre planes
- Sixteen-dimensional compact translation planes with automorphism groups of dimension at least 35
