Advertisement

Real Hypersurfaces in the Complex Hyperbolic Quadric with Commuting Ricci Tensor

  • Young Jin Suh
  • Doo Hyun Hwang
Article
  • 27 Downloads

Abstract

We study Ricci-commuting real hypersurfaces in the complex hyperbolic quadric \({{Q^m}^*} = SO^{o}_{2,m}/SO_mSO_2\), \(m \ge 3\). The commuting Ricci tensor shows that the unit normal vector field N of \({Q^m}^*\) is singular, that is, \(AN=N\) or \(N=\frac{1}{\sqrt{2}}(Z_1+JZ_2)\), \(Z_1,Z_2 \in V(A)\) for a complex conjugation \(A \in {\mathfrak {A}}\), where \(\mathfrak {A}\) denotes the set of all complex conjugations in \({Q^m}^*\). Then according to each case, we give a classification of real hypersurfaces having a commuting Ricci tensor in \({Q^m}^*\).

Keywords

Commuting Ricci tensor \(\mathfrak {A}\)-isotropic \(\mathfrak {A}\)-principal Kähler structure Complex conjugation Complex hyperbolic quadric 

Mathematics Subject Classification

Primary 53C40 Secondary 53C55 

Notes

Acknowledgements

The present authors would like to express their deep gratitude to the referee for his/her careful reading of our article and valuable suggestions to improve the first version of this manuscript. Funding was provided by National Research Foundation of Korea (Grant No. NRF-2018-R1D1A1B-05040381).

References

  1. 1.
    Berndt, J., Suh, Y.J.: Contact hypersurfaces in Kaehler manifold. Proc. Am. Math. Soc. 143, 2637–2649 (2015)CrossRefGoogle Scholar
  2. 2.
    Eberlein, P.B.: Geometry of Nonpositively Curved Manifolds. University of Chicago Press, Chicago (1996)zbMATHGoogle Scholar
  3. 3.
    Hutching, M., Taubes, C.H.: The Weinstein conjecture for stable Hamiltonian structures. Geom. Topol. 13, 901–941 (2009)MathSciNetCrossRefGoogle Scholar
  4. 4.
    Klein, S.: Totally geodesic submanifolds in the complex quadric. Differ. Geom. Appl. 26, 79–96 (2008)MathSciNetCrossRefGoogle Scholar
  5. 5.
    Klein, S., Suh, Y.J.: Contact real hypersurfaces in the complex hyperbolic quadric. SubmittedGoogle Scholar
  6. 6.
    Knap, A.W.: Lie Groups Beyond an Introduction, Progress in Mathematics. Birkhäuser, Basel (2002)Google Scholar
  7. 7.
    Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, Wiley Classics Library Edition, vol. II. Wiley-Interscience Publication, New York (1996)Google Scholar
  8. 8.
    Reckziegel, H.: On the geometry of the complex quadric. In: Dillen, F., Komrakov, B., Simon, U., Van de Woestyne, I., Verstraelen, L. (eds.) Geometry and Topology of Submanifolds VIII (Brussels/Nordfjordeid 1995), pp. 302–315. World Science Publications, River Edge (1995)Google Scholar
  9. 9.
    Smyth, B.: Homogeneous complex hypersurfaces. J. Math. Soc. Jpn. 20, 643–647 (1968)MathSciNetCrossRefGoogle Scholar
  10. 10.
    Smyth, B.: Differential geometry of complex hypersurfaces. Ann. Math. 85, 246–266 (1967)MathSciNetCrossRefGoogle Scholar
  11. 11.
    Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with commuting Ricci tensor. J. Geom. Phys. 60, 1792–1805 (2010)MathSciNetCrossRefGoogle Scholar
  12. 12.
    Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor. Proc. R. Soc. Edinb. A 142, 1309–1324 (2012)MathSciNetCrossRefGoogle Scholar
  13. 13.
    Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with harmonic curvature. J. Math. Pures Appl. 100, 16–33 (2013)MathSciNetCrossRefGoogle Scholar
  14. 14.
    Suh, Y.J.: Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians. Adv. Appl. Math. 50, 645–659 (2013)MathSciNetCrossRefGoogle Scholar
  15. 15.
    Suh, Y.J.: Real hypersurfaces in the complex quadric with Reeb parallel shape operator. Int. J. Math. 25, 1450059 (2014). 17ppMathSciNetCrossRefGoogle Scholar
  16. 16.
    Suh, Y.J.: Real hypersurfaces in the complex hyperbolic two-plane Grassmannians with commuting Ricci tensor. Int. J. Math. 26, 1550008 (2015). 26ppMathSciNetCrossRefGoogle Scholar
  17. 17.
    Suh, Y.J.: Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. Math. 281, 886–905 (2015)MathSciNetCrossRefGoogle Scholar
  18. 18.
    Suh, Y.J.: Real hypersurfaces in complex quadric with harmonic curvature. J. Math. Pures Appl. 106, 393–410 (2016)MathSciNetCrossRefGoogle Scholar
  19. 19.
    Suh, Y.J.: Contact real hypersurfaces in complex hyperbolic quadric. SubmittedGoogle Scholar
  20. 20.
    Suh, Y.J.: Real hypersurfaces in the complex hyperbolic quadric with isometric Reeb flow. Commun. Contemp. Math. 20, 1750031 (2018). (20 pages)MathSciNetCrossRefGoogle Scholar
  21. 21.
    Suh, Y.J., Hwang, D.H.: Real hypersurfaces in the complex quadric with commuting Ricci tensor. SCI China Math. 59, 2185–2198 (2016)MathSciNetCrossRefGoogle Scholar
  22. 22.
    Suh, Y.J., Hwang, D.H.: Real hypersurfaces in the complex hyperbolic quadric with Reeb parallel shape operator. Ann. Mat. Pura Appl. 196, 1307–1326 (2017)MathSciNetCrossRefGoogle Scholar
  23. 23.
    Suh, Y.J., Woo, C.: Real hypersurfaces in complex hyperbolic two-plane Grassmannians with parallel Ricci tensor. Math. Nachr. 287, 1524–1529 (2014)MathSciNetCrossRefGoogle Scholar

Copyright information

© Malaysian Mathematical Sciences Society and Penerbit Universiti Sains Malaysia 2018

Authors and Affiliations

  1. 1.Department of Mathematics and RIRCM, College of Natural SciencesKyungpook National UniversityDaeguRepublic of Korea

Personalised recommendations