Comptes Rendus
no. 7
Differential geometry
Conformally flat real hypersurfaces in nonflat complex planes
[Hypersurfaces réelles conformément plates dans les plans complexes non plats]

Présenté par : the Editorial Board

Yaning Wang1
1 School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, PR China
Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 823-829.

Dans cette note, nous démontrons qu'il n'existe pas d'hypersurface réelle conformément plate dans les espaces de formes complexes de dimension deux, non plats, pourvu que le champ de vecteurs structurel soit champ de vecteur propre de l'opérateur de Ricci. Ceci étend des résultats récents de Cho (Conformally flat normal almost contact 3-manifolds, Honam Math. J. 38 (2016) 59–69) et Kron (3-dimensional real hypersurfaces with η-harmonic curvature, in : Hermitian–Grassmannian Submanifolds, Springer, Singapore, 2017, pp. 155–164).

In this paper, we prove that there are no conformally flat real hypersurfaces in nonflat complex space forms of complex dimension two provided that the structure vector field is an eigenvector field of the Ricci operator. This extends some recent results by Cho (Conformally flat normal almost contact 3-manifolds, Honam Math. J. 38 (2016) 59–69) and Kon (3-dimensional real hypersurfaces with η-harmonic curvature, in: Hermitian–Grassmannian Submanifolds, Springer, Singapore, 2017, pp. 155–164).

Reçu le :
Accepté le :
Publié le :
DOI : 10.1016/j.crma.2018.04.025
Yaning Wang 1

1 School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, PR China 
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Yaning Wang. Conformally flat real hypersurfaces in nonflat complex planes. Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 823-829. doi : 10.1016/j.crma.2018.04.025. https://comptes-rendus.academie-sciences.fr/mathematique/articles/10.1016/j.crma.2018.04.025/

[1] A. Besse Einstein Manifolds, Springer-Verlag, 1987

[2] G. Calvaruso; D. Perrone; L. Vanhecke Homogeneity on three-dimensional contact metric manifolds, Isr. J. Math., Volume 114 (1999), pp. 301-321

[3] J.T. Cho Conformally flat normal almost contact 3-manifolds, Honam Math. J., Volume 38 (2016), pp. 59-69

[4] P. Dacko; Z. Olszak On conformally almost cosymplectic manifolds with Kählerian leaves, Rend. Semin. Mat. Univ. Politec. Torino, Volume 56 (1998), pp. 89-103

[5] A. Derdziński Classification of certain compact Riemannian manifolds with harmonic curvature and non-parallel Ricci tensor, Math. Z., Volume 172 (1980), pp. 273-280

[6] U.H. Ki Real hypersurfaces with parallel Ricci tensor of a complex space form, Tsukuba J. Math., Volume 13 (1989), pp. 73-81

[7] U.H. Ki; H.J. Kim; H. Nakagawa Real hypersurfaces with η-parallel Weyl tensor of a complex space from, J. Korean Math. Soc., Volume 26 (1989), pp. 311-322

[8] U.H. Ki; S. Nagai Real hypersurfaces of a nonflat complex space form in terms of the Ricci tensor, Tsukuba J. Math., Volume 29 (2005), pp. 511-532

[9] U.H. Ki; H. Nakagawa; Y.J. Suh Real hypersurfaces with harmonic Weyl tensor of a complex space form, Hiroshima Math. J., Volume 20 (1990), pp. 93-102

[10] H.J. Kim A note on real hypersurfaces of a complex hyperbolic space, Tsukuba J. Math., Volume 12 (1988), pp. 451-457

[11] M. Kimura Real hypersurfaces of a complex projective space, Bull. Aust. Math. Soc., Volume 33 (1986), pp. 383-387

[12] M. Kon 3-dimensional real hypersurfaces and Ricci operator, Differ. Geom. Dyn. Syst., Volume 16 (2014), pp. 189-202

[13] M. Kon Ricci tensor of real hypersurfaces, Pac. J. Math., Volume 281 (2016), pp. 103-123

[14] M. Kon 3-dimensional real hypersurfaces with η-harmonic curvature, Hermitian–Grassmannian Submanifolds, Springer, Singapore, 2017, pp. 155-164

[15] J.H. Kwon; H. Nakagawa A note on real hypersurfaces of a complex projective space, J. Aust. Math. Soc., Volume 47 (1989), pp. 108-113

[16] C. Li; U.H. Ki Structure eigenvectors of the Ricci tensor in a real hypersurface of a complex projective space, Kyungpook Math. J., Volume 46 (2006), pp. 463-476

[17] Y. Maeda On real hypersurfaces of a complex projective space, J. Math. Soc. Jpn., Volume 28 (1976), pp. 529-540

[18] R. Niebergall; P.J. Ryan Real hypersurfaces in complex space forms, Tight and Taut Submanifolds, Math. Sci. Res. Inst. Publ., vol. 32, Cambridge University Press, Cambridge, UK, 1997, pp. 233-305

[19] S. Nishikawa; Y. Maeda Conformally flat hypersurfaces in a conformally flat Riemannian manifold, Tohoku Math. J., Volume 26 (1974), pp. 159-168

[20] K. Panagiotidou The structure Jacobi operator and the shape operator of real hypersurfaces in CP2 and CH2, Beitr. Algebra Geom., Volume 55 (2014), pp. 545-556

[21] K. Panagiotidou; P.J. Xenos Real hypersurfaces in CP2 and CH2 whose structure Jacobi operator is Lie D-parallel, Note Mat., Volume 32 (2012), pp. 89-99

[22] Y. Wang Conformally flat almost Kenmotsu 3-manifolds, Mediterr. J. Math., Volume 14 (2017), p. 186

[23] Y. Wang Cotton tensors on almost coKähler 3-manifolds, Ann. Pol. Math., Volume 120 (2017), pp. 135-148

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