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Non-existence of conformally flat real hypersurfaces in both the complex quadric and the complex hyperbolic quadric

Part of: Global differential geometry Local differential geometry

Published online by Cambridge University Press:  15 February 2021

Zeke Yao ,
Bangchao Yin  and
Zejun Hu
Zeke Yao
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou450001, P.R. China e-mail: yaozkleon@163.commathyinchao@163.com
Bangchao Yin
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou450001, P.R. China e-mail: yaozkleon@163.commathyinchao@163.com
Zejun Hu*
Affiliation:
School of Mathematics and Statistics, Zhengzhou University, Zhengzhou450001, P.R. China e-mail: yaozkleon@163.commathyinchao@163.com
*
Zejun Hu is the corresponding author. e-mail: huzj@zzu.edu.cn
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Abstract

In this paper, by applying for a new approach of the so-called Tsinghua principle, we prove the nonexistence of locally conformally flat real hypersurfaces in both the m-dimensional complex quadric $Q^m$ and the complex hyperbolic quadric $Q^{m\ast }$ for $m\ge 3$ .

Keywords

Complex quadriccomplex hyperbolic quadricconformally flat hypersurfacereal hypersurface𝔄-principalTsinghua principle

MSC classification

Primary: 53B25: Local submanifolds
Secondary: 53C40: Global submanifolds 53C55: Hermitian and Kählerian manifolds
Type
Article
Information
Canadian Mathematical Bulletin , Volume 65 , Issue 1 , March 2022 , pp. 68 - 83
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Copyright
© Canadian Mathematical Society 2021

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Footnotes

This project was supported by NSF of China, Grant Number 11771404.

References

Antić, M., Li, H., Vrancken, L., and Wang, X., Affine hypersurfaces with constant sectional curvature. Pacific J. Math. 310(2021), 275–302. https://protect-eu.mimecast.com/s/1bBJCK8y8fZW96VtMUMmA?domain=mathscinet.ams.org 10.2140/pjm.2021.310.275CrossRefGoogle Scholar
Berndt, J. and Suh, Y. J., On the geometry of homogeneous real hypersurfaces in the complex quadric. In: Proceedings of the 16th International Workshop on Differential Geometry and the 5th KNUGRG-OCAMI Differential Geometry Workshop, vol. 16, 2012, pp. 1–9.Google Scholar
Berndt, J. and Suh, Y. J., Real hypersurfaces with isometric Reeb flow in complex quadrics. Int. J. Math. 24(2013), Article no. 1350050, 18 pp.10.1142/S0129167X1350050XCrossRefGoogle Scholar
Cartan, E., La déformation des hypersurfaces dans l’espace conforme réel à $n\ge 5$ dimensions. French. Bull. Soc. Math. Fr. 45(1917), 57–121.10.24033/bsmf.975CrossRefGoogle Scholar
Cecil, T. E. and Ryan, P. J., Focal sets and real hypersurfaces in complex projective space. Trans. Amer. Math. Soc. 269(1982), 481–499.Google Scholar
Cheng, X., Hu, Z., Moruz, M., and Vrancken, L., On product minimal Lagrangian submanifolds in complex space forms. J. Geom. Anal. 31 (2021), 1934–1964. http://dx.doi.org/10.1007/s12220-019-00328-7 CrossRefGoogle Scholar
Cheng, X., Hu, Z., Moruz, M., and Vrancken, L., On product affine hyperspheres in ℝn+1 . Sci. China Math. 63(2020), 2055–2078.10.1007/s11425-018-9457-9CrossRefGoogle Scholar
Dioos, B., Vrancken, L., and Wang, X., Lagrangian submanifolds in the homogeneous nearly Kähler S3 × S3 . Ann. Global Anal. Geom. 53(2018), 39–66.10.1007/s10455-017-9567-zCrossRefGoogle Scholar
do Carmo, M., Dajczer, M., and Mercuri, F., Compact conformally flat hypersurfaces. Trans. Amer. Math. Soc. 288(1985), 189–203.10.1090/S0002-9947-1985-0773056-0CrossRefGoogle Scholar
Hu, Z. and Yin, J., New characterizations of real hypersurfaces with isometric Reeb flow in the complex quadric. Colloq. Math. 164 (2021), 211–219. http://dx.doi.org/10.4064/cm8075-12-2019CrossRefGoogle Scholar
Ki, U.-H., Nakagawa, H., and Suh, Y. J., Real hypersurfaces with harmonic Weyl tensor of a complex space form. Hiroshima Math. J. 20(1990), 93–102.10.32917/hmj/1206454442CrossRefGoogle Scholar
Klein, S., Totally geodesic submanifolds of the complex quadric. Differential Geom. Appl. 26(2008), 79–96.CrossRefGoogle Scholar
Klein, S. and Suh, Y. J., Contact real hypersurfaces in the complex hyperbolic quadric. Ann. Mat. Pura Appl. 198(2019), 1481–1494.10.1007/s10231-019-00827-yCrossRefGoogle Scholar
Kon, M., Pseudo-Einstein real hypersurfaces in complex space forms. J. Differential Geom. 14(1979), 339–354.10.4310/jdg/1214435100CrossRefGoogle Scholar
Lee, H. and Suh, Y. J., Commuting Jacobi operators on real hypersurfaces of type B in the complex quadric. Math. Phys. Anal. Geom. 23(2020), Article no. 44.CrossRefGoogle Scholar
Li, H., Ma, H., Van der Veken, J., Vrancken, L., and Wang, X., Minimal Lagrangian submanifolds of the complex hyperquadric. Sci. China Math. 63(2020), 1441–1462.10.1007/s11425-019-9551-2CrossRefGoogle Scholar
Loo, T. H., 𝔄-principal Hopf hypersurfaces in complex quadrics. arxiv:1712.00538v1 Google Scholar
Montiel, S., Real hypersurfaces of a complex hyperbolic space. J. Math. Soc. Jpn. 37(1985), 515–535.CrossRefGoogle Scholar
Montiel, S. and Romero, A., Complex Einstein hypersurfaces of indefinite complex space forms. Math. Proc. Cambridge Philos. Soc. 94(1983), 495–508.10.1017/S0305004100000888CrossRefGoogle Scholar
Nishikawa, S. and Maeda, Y., Conformally flat hypersurfaces in a conformally flat Riemannian manifold. Tohoku Math. J. 26(1974), 159–168.10.2748/tmj/1178241240CrossRefGoogle Scholar
Pinkall, U., Compact conformally flat hypersurfaces. In: Conformal geometry (Bonn, 1985/1986), Aspects Math., E12, Friedr. Vieweg, Braunschweig, 1988, pp. 217–236.CrossRefGoogle Scholar
Reckziegel, H., On the geometry of the complex quadric. In: Geometry and topology of submanifolds, VIII (Brussels, 1995/Nordfjordeid, 1995), World Sci. Publ., River Edge, NJ, 1996, pp. 302–315.Google Scholar
Smyth, B., Differential geometry of complex hypersurfaces. Ann. Math. 85(1967), 246–266.CrossRefGoogle Scholar
Suh, Y. J., Real hypersurfaces in the complex quadric with harmonic curvature. J. Math. Pures Appl. 106(2016), 393–410.CrossRefGoogle Scholar
Suh, Y. J., Real hypersurfaces in the complex hyperbolic quadrics with isometric Reeb flow. Commun. Contemp. Math. 20(2018), Article no. 1750031, 20 pp.CrossRefGoogle Scholar
Suh, Y. J. and Hwang, D. H., Real hypersurfaces in the complex quadric with commuting Ricci tensor. Sci. China Math. 59(2016), 2185–2198.CrossRefGoogle Scholar
Suh, Y. J., Pérez, J. D., and Woo, C., Real hypersurfaces in the complex hyperbolic quadric with parallel structure Jacobi operator. Publ. Math. Debrecen 94(2019), 75–107.10.5486/PMD.2019.8262CrossRefGoogle Scholar
Van der Veken, J. and Wijffels, A., Lagrangian submanifolds of the complex hyperbolic quadric. Ann. Mat. Pura Appl. https://doi.org/10.1007/s10231-020-01063-5 arxiv:2002.10314v1 CrossRefGoogle Scholar
Hu, Z., Moruz, M., Vrancken, L. and Yao, Z., On the nonexistence and rigidity for hypersurfaces of the homogeneous nearly Kȁhler $S^3\times S^3$ , Differential Geom. Appl. 75 (2021), Article no. 101717.CrossRefGoogle Scholar