Hostname: page-component-745bb68f8f-hvd4g Total loading time: 0 Render date: 2025-01-11T12:01:09.600Z Has data issue: false hasContentIssue false
  • English
  • Français

A new classification on parallel Ricci tensor for real hypersurfaces in the complex quadric

Part of: Global differential geometry

Published online by Cambridge University Press:  18 November 2020

Hyunjin Lee  and
Young Jin Suh
Hyunjin Lee
Affiliation:
The Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu41566, Republic of Korea (lhjibis@hanmail.net)
Young Jin Suh
Affiliation:
Kyungpook National University, College of Natural Sciences, Department of Mathematics and Research Institute of Real & Complex Manifolds Daegu 41566, Republic of Korea (yjsuh@knu.ac.kr)
Get access Rights & Permissions [Opens in a new window]

Abstract

First we introduce the notion of parallel Ricci tensor ${\nabla }\mathrm {Ric}=0$ for real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 and show that the unit normal vector field N is singular. Next we give a new classification of real hypersurfaces in the complex quadric Qm = SOm+2/SOmSO2 with parallel Ricci tensor.

Keywords

Parallel Ricci tensor$\mathfrak {A}$-isotropic$\mathfrak {A}$-principalKähler structurecomplex conjugationcomplex quadric

MSC classification

Primary: 53C40: Global submanifolds
Secondary: 53C55: Hermitian and Kählerian manifolds
Type
Research Article
Information
Proceedings of the Royal Society of Edinburgh Section A: Mathematics , Volume 151 , Issue 6 , December 2021 , pp. 1846 - 1868
Check for updates
Copyright
Copyright © The Author(s), 2020. Published by Cambridge University Press on behalf of The Royal Society of Edinburgh

Access options

Get access to the full version of this content by using one of the access options below. (Log in options will check for institutional or personal access. Content may require purchase if you do not have access.)

References

Berndt, J. and Suh, Y. J.. Real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 127 (1999), 1–14.CrossRefGoogle Scholar
Berndt, J. and Suh, Y. J.. Isometric flows on real hypersurfaces in complex two-plane Grassmannians. Monatsh. Math. 137 (2002), 87–98.10.1007/s00605-001-0494-4CrossRefGoogle Scholar
Berndt, J. and Suh, Y. J.. Real hypersurfaces with isometric Reeb flow in complex quadrics. Int. J. Math. 24 (2013), 1350050, 18pp.10.1142/S0129167X1350050XCrossRefGoogle Scholar
Berndt, J. and Suh, Y. J.. Contact hypersurfaces in Kaehler manifold. Proc. Am. Math. Soc. 143 (2015), 2637–2649.CrossRefGoogle Scholar
Berndt, J. and Suh, Y. J.. Real hypersurfaces in Hermitian symmetric spaces. In Advances in Analysis and Geometry (Editor in Chief, Xiao, J.). ©2021 Copyright-Text. (Berlin/Boston: Walter de Gruyter GmbH, in press).Google Scholar
Besse, A. L.. Einstein Manifolds, Reprint of the 1987 edition, Classics in Mathematics, (Berlin: Springer-Verlag, 2008).Google Scholar
Blair, D. E.. Almost contact manifolds with Killing structure tensors. Pacific J. Math. 39 (1971), 285–292.CrossRefGoogle Scholar
Helgason, S.. Differential geometry, Lie Groups and symmetric spaces, graduate studies in math. Am. Math. Soc. 34 (2001).Google Scholar
Kimura, M.. Real hypersurfaces of a complex projective space. Bull. Aust. Math. Soc. 33 (1986), 383–387.CrossRefGoogle Scholar
Klein, S.. Totally geodesic submanifolds in the complex quadric. Differential Geom. Appl. 26 (2008), 79–96.10.1016/j.difgeo.2007.11.004CrossRefGoogle Scholar
Knapp, A. W.. Lie Groups Beyond an Introduction, Second edition, Progress in Mathematics, 140 (Boston, MA: Birkhäuser Boston, Inc., 2002).Google Scholar
Kobayashi, S. and Nomizu, K.. Foundations of Differential Geometry, Vol. II, Reprint of the 1969 original (New York: Wiley Classics Library, A Wiley-Interscience Publication, John Wiley & Sons, Inc., 1996).Google Scholar
Kwon, J-H. and Nakagawa, H.. Real hypersurfaces with cyclic-parallel Ricci tensor of a complex projective space. Hokkaido Math. J. 17 (1988), 355–371.10.14492/hokmj/1381517819CrossRefGoogle Scholar
Lee, H. and Suh, Y.J.. Commuting Jacobi operators on Real hypersurfaces of Type $\rm B$ in the complex quadric. ArXiv:2003.07231 [math.DG].Google Scholar
Lee, H., Hwang, D. and Suh, Y. J.. Real hypersurfaces in the complex quadric with generalized Killing shape operator. J. Geom. Phys. (in press 2021), https://doi.org/10.1016/j.geomphys.2020.103800.Google Scholar
Montiel, S. and Romero, A.. On some real hypersurfaces in a complex hyperbolic space. Geom. Dedicata 212 (1991), 355–364.Google Scholar
Okumura, M.. On some real hypersurfaces of a complex projective space. Trans. Am. Math. Soc. 212 (1975), 355–364.CrossRefGoogle Scholar
Pérez, J. D.. Cyclic-parallel real hypersurfaces of quaternionic projective space. Tsukuba J. Math. 17 (1993), 189–191.CrossRefGoogle Scholar
Pérez, J. D. and Suh, Y. J.. Real hypersurfaces of quaternionic projective space satisfying ${\nabla }_{U_i}R=0$. Differential Geom. Appl. 7 (1997), 211–217.10.1016/S0926-2245(97)00003-XCrossRefGoogle Scholar
Pérez, J. D. and Suh, Y. J.. The Ricci tensor of real hypersurfaces in complex two-plane Grassmannians. J. Korean Math. Soc. 44 (2007), 211–235.10.4134/JKMS.2007.44.1.211CrossRefGoogle Scholar
Pérez, J. D. and Suh, Y. J.. Derivatives of the shape operator of real hypersurfaces in the complex quadric. Results Math. 73 (2018), 126.CrossRefGoogle Scholar
Reckziegel, H.. On the geometry of the complex quadric. In: Geometry and Topology of Submanifolds VIII (Brussels/Nordfjordeid 1995), pp. 302–315 (River Edge, NJ: World Sci. Publ., 1995).Google Scholar
Smyth, B.. Differential geometry of complex hypersurfaces. Ann. Math. 85 (1967), 246–266.10.2307/1970441CrossRefGoogle Scholar
Suh, Y. J.. Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor. Proc. Roy. Soc. Edinb. A 142 (2012), 1309–1324.10.1017/S0308210510001472CrossRefGoogle Scholar
Suh, Y. J.. Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians. Adv. Appl. Math. 50 (2013), 645–659.CrossRefGoogle Scholar
Suh, Y. J.. Real hypersurfaces in the complex quadric with Reeb parallel shape operator. Int. J. Math. 25 (2014), 1450059, 17pp.CrossRefGoogle Scholar
Suh, Y. J.. Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. Math. 281 (2015), 886–905.10.1016/j.aim.2015.05.012CrossRefGoogle Scholar
Suh, Y. J.. Pseudo-Einstein real hypersurfaces in the complex quadric. Math. Nachr. 290 (2017), 1884–1904.CrossRefGoogle Scholar
Suh, Y. J.. Generalized Killing Ricci tensor for real hypersurfaces in the complex two-plane Grassmannians. J. Geom. Phys. (in press 2021), https://doi.org/10.1016/j.geophys.2020.103799.Google Scholar
Suh, Y. J.. Generalized Killing Ricci tensor for real hypersurfaces in the complex hyperbolic two-plane Grassmannians. Mediterr. J. Math. (in press 2021).CrossRefGoogle Scholar
Yano, K.. On harmonic and Killing vectors. Ann. Math. 55 (1952), 38–45.10.2307/1969418CrossRefGoogle Scholar
Yano, K.. Harmonic and Killing tensor fields in Riemannian spaces with boundary. J. Math. Soc. Japan 10 (1958), 43–437.10.2969/jmsj/01040430CrossRefGoogle Scholar
Yano, K.. Harmonic and Killing tensor fields in compact orientable Riemannian spaces with boundary. Ann. Math. 69 (1959), 588–597.CrossRefGoogle Scholar