Differential geometry
Conformally flat real hypersurfaces in nonflat complex planes
Comptes Rendus. Mathématique, Volume 356 (2018) no. 7, pp. 823-829.

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).

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).

Received:
Accepted:
Published online:
DOI: 10.1016/j.crma.2018.04.025
Wang, Yaning 1

1 School of Mathematics and Information Sciences, Henan Normal University, Xinxiang 453007, Henan, PR China
@article{CRMATH_2018__356_7_823_0,
     author = {Wang, Yaning},
     title = {Conformally flat real hypersurfaces in nonflat complex planes},
     journal = {Comptes Rendus. Math\'ematique},
     pages = {823--829},
     publisher = {Elsevier},
     volume = {356},
     number = {7},
     year = {2018},
     doi = {10.1016/j.crma.2018.04.025},
     language = {en},
     url = {http://www.numdam.org/articles/10.1016/j.crma.2018.04.025/}
}
TY  - JOUR
AU  - Wang, Yaning
TI  - Conformally flat real hypersurfaces in nonflat complex planes
JO  - Comptes Rendus. Mathématique
PY  - 2018
SP  - 823
EP  - 829
VL  - 356
IS  - 7
PB  - Elsevier
UR  - http://www.numdam.org/articles/10.1016/j.crma.2018.04.025/
DO  - 10.1016/j.crma.2018.04.025
LA  - en
ID  - CRMATH_2018__356_7_823_0
ER  - 
%0 Journal Article
%A Wang, Yaning
%T Conformally flat real hypersurfaces in nonflat complex planes
%J Comptes Rendus. Mathématique
%D 2018
%P 823-829
%V 356
%N 7
%I Elsevier
%U http://www.numdam.org/articles/10.1016/j.crma.2018.04.025/
%R 10.1016/j.crma.2018.04.025
%G en
%F CRMATH_2018__356_7_823_0
Wang, Yaning. 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. http://www.numdam.org/articles/10.1016/j.crma.2018.04.025/

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

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

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

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

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

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

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

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

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

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

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

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

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

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

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

[16] Li, C.; Ki, U.H. 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] Maeda, Y. On real hypersurfaces of a complex projective space, J. Math. Soc. Jpn., Volume 28 (1976), pp. 529-540

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

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

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

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

Cited by Sources: