Abstract
Let M be a real hypersurface in nonflat complex space forms of complex dimension two. In this paper, we prove that the shape operator of M is transversally Killing with respect to the generalized Tanaka-Webster connection if and only if M is locally congruent to a type (A) or (B) real hypersurface. We also prove that shape operator of M commutes with Cho operator on holomorphic distribution if and only if M is locally congruent to a ruled real hypersurface.
This work was supported by the National Natural Science Foundation of China (No. 11431009).
(Communicated by Július Korbaš)
Acknowledgement
The authors would like to thank the referee for many useful comments which improves the original paper.
References
[1] Ahn, S. S.—Lee, S. B.—Suh, Y. J.: On ruled real hypersurfaces in a complex space form, Tsukuba J. Math. 17 (1993), 311–322.10.21099/tkbjm/1496162264Suche in Google Scholar
[2] Berndt, J.: Real hypersurfaces with constant principal curvatures in complex hyperbolic space, J. Reine Angew. Math. 395 (1989), 132–141.10.1112/S0024610706023295Suche in Google Scholar
[3] Blair, D. E.: Almost contact manifolds with Killing structure tensors, Pacific J. Math. 39 (1971), 285–292.10.2140/pjm.1971.39.285Suche in Google Scholar
[4] Cho, J. T.: CR-structures on real hypersurfaces of a complex space form, Publ. Math. Debrecen 54 (1999), 473–487.10.5486/PMD.1999.2081Suche in Google Scholar
[5] Cho, J. T.: Pseudo-Einstein CR-structures on real hypersurfaces in a complex space form, Hokkaido Math. J. 37 (2008), 1–17.10.14492/hokmj/1253539581Suche in Google Scholar
[6] Cho, J. T.: Notes on real hypersurfaces in a complex space form, Bull. Korean Math. Soc. 52 (2015), 335–344.10.4134/BKMS.2015.52.1.335Suche in Google Scholar
[7] Cho, J. T.—Kimura, M.: Transversal symmetries on real hypersurfaces in a complex space form, Hiroshima Math. J. 43 (2013), 223–238.10.32917/hmj/1372180513Suche in Google Scholar
[8] Kim, H. S.—Pyo, Y. S.: On real hypersurfaces of type A in a complex space form (III), Balkan J. Geom. Appl. 3 (1998), 101–110.Suche in Google Scholar
[9] Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space, Trans. Amer. Math. Soc. 296 (1986), 137–149.10.1090/S0002-9947-1986-0837803-2Suche in Google Scholar
[10] Kimura, M.: Sectional curvatures of holomorphic planes on a real hypersurfaces in Pn(ℂ), Math. Ann. 276 (1987), 487–497.10.1007/BF01450843Suche in Google Scholar
[11] Kimura, M.—Maeda, S.: On real hypersurfaces of a complex projective space, Math. Z. 202 (1989), 299–311.10.1007/BF01159962Suche in Google Scholar
[12] Kon, M.: Real hypersurfaces in a 2-dimensional complex space form with transversal Killing vector fields, Proc. Nineteenth International Workshop on Hermitian-Grassmannian Submanifolds, 19 (2015), 159–170.Suche in Google Scholar
[13] Kon, S. H.—Loo, T. H.: Real hypersurfaces in a complex space form with η-parallel shape operator, Math. Z. 269 (2011), 47–58.10.1007/s00209-010-0715-4Suche in Google Scholar
[14] Maeda, Y.: On real hypersurfaces of a complex projective space, J. Math. Soc. Japan 28 (1976), 529–540.10.2969/jmsj/02830529Suche in Google Scholar
[15] Montiel, S.: Real hypersurfaces of a complex hyperbolic space, J. Math. Soc. Japan 37 (1985), 515–535.10.2969/jmsj/03730515Suche in Google Scholar
[16] Montiel, S.—Romero, A.: On some real hypersurfaces of a complex hyperbolic space, Geom. Dedicata 20 (1986), 245–261.10.1007/BF00164402Suche in Google Scholar
[17] Niebergall, R.—Ryan, P. J.: Real hypersurfaces in complex space forms. In: Tight and Taut Submanifolds, Math. Sci. Res. Inst. Publ. 32, Cambridge Univ. Press, Cambridge, 1997.Suche in Google Scholar
[18] Okumura, M.: On some real hypersurfaces of a complex projective space, Trans. Amer. Math. Soc. 212, (1975), 355–364.10.1090/S0002-9947-1975-0377787-XSuche in Google Scholar
[19] Panagiotidou, K.—Pérez, J. D.: Commuting conditions of the k-th Cho operator with the structure Jacobi operator of real hypersurfaces in complex space forms, Open Math. 13 (2015), 321–332.10.1515/math-2015-0032Suche in Google Scholar
[20] Panagiotidou, K.—Xenos, P. J.: Real hypersurfaces in ℂP2and ℂH2whose structure Jacobi operator is Lie 𝔻-parallel, Note Mat. 32 (2012), 89–99.Suche in Google Scholar
[21] Pérez, J. D.—Suh, Y. J.: A characterization of type (A) real hypersurfaces in complex projective space, Publ. Math. Debrecen 83 (2013), 707–714.10.5486/PMD.2013.5712Suche in Google Scholar
[22] Pérez, J. D.: A characterization of type A ral hypersurfaces in complex projective space, Publ. Math. Debreceen 84 (2013), 707–714.10.5486/PMD.2013.5712Suche in Google Scholar
[23] Pérez, J. D.: Commutativity of Cho and structure Jacobi operators of a real hypersurface in a complex projective space, Ann. Mat. Pura Appl. 194 (2015), 1781–1794.10.1007/s10231-014-0444-0Suche in Google Scholar
[24] Pérez, J. D.—Suh, Y. J.: Generalized Tanaka-Webster and covariant derivatives on a real hypersurface in a complex projective space, Monatsh Math. 177 (2015), 637–647.10.1007/s00605-015-0777-9Suche in Google Scholar
[25] Takagi, R.: Real hypersurfaces in complex projective space with constant principal curvatures, J. Math. Soc. Japan 27 (1975), 43–53.10.2969/jmsj/02710043Suche in Google Scholar
[26] Takagi, R.: Real hypersurfaces in complex projective space with constant principal curvatures II, J. Math. Soc. Japan 27 (1975), 507–516.10.2969/jmsj/02740507Suche in Google Scholar
[27] Tanaka, N.: On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections, Jpn. J. Math. 2 (1976), 131–190.10.4099/math1924.2.131Suche in Google Scholar
[28] Tanno, S.: Variational problems on contact Riemannian manifolds, Trans. Amer. Math. Soc. 314 (1989), 349–379.10.1090/S0002-9947-1989-1000553-9Suche in Google Scholar
[29] Webster, S. M.: Pseudohermitian structures on a real hypersurface, J. Differential Geom. 13 (1978), 25–41.10.4310/jdg/1214434345Suche in Google Scholar
[30] Wang, Y.: Real hypersurfaces with Killing type operators in a nonflat complex space form, J. Geom. 108 (2017), 825–835.10.1007/s00022-017-0375-1Suche in Google Scholar
© 2019 Mathematical Institute Slovak Academy of Sciences
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Artikel in diesem Heft
- Regular papers
- The life jubilee of Prof. RNDr. Sylvia Pulmannová, DrSc.
- Perfect 1-factorizations
- A topological duality for strong Boolean posets
- On the Diophantine equations x2 + 2α 3β 19γ = yn and x2 + 2α 3β 13γ = yn
- Tribonacci numbers and primes of the form p = x2 + 11y2
- Basic semirings
- A conjecture for varieties of completely regular semigroups
- Uniqueness of meromorphic function with its shift operator under the purview of two or three shared sets
- Differential subordination results for Mittag-Leffler type functions with bounded turning property
- Mittag-Leffler stability for non-instantaneous impulsive Caputo fractional differential equations with delays
- Asymptotically periodic behavior of solutions of fractional evolution equations of order 1 < α < 2
- On the polynomial entropy for morse gradient systems
- Quantitative approximation by Stancu-Durrmeyer-Choquet-Šipoš operators
- A note on non-linear ∗-Jordan derivations on ∗-algebras
- Disjoint hypercyclic weighted translations on locally compact hausdorff spaces
- Some new results on real hypersurfaces with generalized Tanaka-Webster connection
- Relative topological properties of hyperspaces
- Cohomology of torus manifold bundles
- The Menger and projective Menger properties of function spaces with the set-open topology
- Asymptotic behavior of the record values in a stationary Gaussian sequence, with applications