Skip to main content
SpringerLink
Log in

Real Hypersurfaces with Killing Shape Operator in the Complex Quadric

  • Published:
Mediterranean Journal of Mathematics Aims and scope Submit manuscript

Abstract

We introduce the notion of Killing shape operator for real hypersurfaces in the complex quadric \(Q^m = SO_{m+2}/SO_mSO_2\). The Killing shape operator condition implies that the unit normal vector field N becomes \(\mathfrak {A}\)-principal or \(\mathfrak {A}\)-isotropic. Then according to each case, we give a complete classification of Hopf real hypersurfaces in \(Q^m = SO_{m+2}/SO_mSO_2\) with Killing shape operator.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

References

  1. Blair, D.E.: Almost contact manifolds with Killing structure tensors. Pac. J. Math. 39, 285–292 (1971)

    Article  MathSciNet  MATH  Google Scholar 

  2. Jeong, I., Kim, H.J., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel normal Jacobi operator. Publ. Math. Debr. 76, 203–218 (2010)

    MathSciNet  MATH  Google Scholar 

  3. Jeong, I., Machado, C.J.G., Pérez, J.D., Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with \(\mathfrak{D}^{\bot }\)-parallel structure Jacobi operator. Int. J. Math. 22, 655–673 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  4. Jeong, I., Machado, C.J.G., Pérez, J.D., Suh, Y.J.: \(\cal{D}\)-parallelism of normal and structure Jacobi operators for hypersurfaces in complex two-plane Grassmannians. Ann. Mat. Pura Appl. 193, 591–608 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  5. Klein, S.: Totally geodesic submanifolds in the complex quadric. Differ. Geom. Appl. 26, 79–96 (2008)

    Article  MATH  Google Scholar 

  6. Klein, S.: Totally geodesic submanifolds of the complex quadric and the quaternionic 2-Grassmannians. Trans. Am. Math. Soc. 361, 4927–4967 (2009)

    Article  MATH  Google Scholar 

  7. Klein, S.: Reconstructing the geometric structure of a Riemannian symmetric space from its Satake diagram. Geom. Dedic. 138, 25–50 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  8. Klein, S.: Totally geodesic submanifolds of the exceptional Riemannian symmetric spaces of rank 2. Osaka J. Math. 47, 1077–1157 (2010)

    MathSciNet  MATH  Google Scholar 

  9. Kimura, M.: Real hypersurfaces and complex submanifolds in complex projective space. Trans. Am. Math. Soc. 296, 137–149 (1986)

    Article  MathSciNet  MATH  Google Scholar 

  10. Kobayashi, S., Nomizu, K.: Foundations of Differential Geometry, vol. II (Wiley classics library edition). Wiley, New York (1996)

  11. Pérez, J.D., Suh, Y.J.: Real hypersurfaces of quaternionic projective space satisfying \({\nabla }_{U_i}R=0\). Differ. Geom. Appl. 7, 211–217 (1997)

    Article  MathSciNet  MATH  Google Scholar 

  12. Pérez, J.D., Suh, Y.J.: Certain conditions on the Ricci tensor of real hypersurfaces in quaternionic projective space. Acta Math. Hung. 91, 343–356 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  13. Reckziegel, H.: On the geometry of the complex quadric. In: Geometry and Topology of Submanifolds VIII. World Scientific Publishing, Brussels/Nordfjordeid, River Edge, pp. 302–315 (1995)

  14. Smyth, B.: Differential geometry of complex hypersurfaces. Ann. Math. 85, 246–266 (1967)

    Article  MathSciNet  MATH  Google Scholar 

  15. Suh, Y.J.: Real hypersurfaces of type B in complex two-plane Grassmannians. Monatsh. Math. 147, 337–355 (2006)

    Article  MathSciNet  MATH  Google Scholar 

  16. Suh, Y.J.: Real hypersurfaces in complex two-plane Grassmannians with parallel Ricci tensor. Proc. R. Soc. Edinb. A. 142, 1309–1324 (2012)

    Article  MathSciNet  MATH  Google Scholar 

  17. Suh, Y.J.: Hypersurfaces with isometric Reeb flow in complex hyperbolic two-plane Grassmannians. Adv. Appl. Math. 50, 645–659 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  18. Suh, Y.J.: Real hypersurfaces in the complex quadric with Reeb parallel shape operator. Int. J. Math. 25, 1450059 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  19. Suh, Y.J.: Real hypersurfaces in the complex quadric with parallel Ricci tensor. Adv. Math. 281, 886–905 (2015)

    Article  MathSciNet  MATH  Google Scholar 

  20. Suh, Y.J.: Real hypersurfaces in the complex quadric with harmonic curvature. J. Math. Pures Appl. 106, 393–410 (2016)

    Article  MathSciNet  MATH  Google Scholar 

  21. Tachibana, S.: On Killing tensors in a Riemannian space. Tohoku Math. J. 20, 257–264 (1968)

    Article  MathSciNet  MATH  Google Scholar 

Download references

Acknowledgements

The present authors would like to express their deep gratitude to the referee for his/her wonderful comments throughout all of our manuscript.

Author information

Authors and Affiliations

  1. Department of Geometry and Topology, University of Granada, 18071, Granada, Spain

    Juan de Dios Pérez

  2. Division of Future Capability Education, Pai Chai University, Daejeon, 35345, Republic of Korea

    Imsoon Jeong

  3. Department of Mathematics and Research Institute of Real and Complex Manifolds, College of Natural Sciences, Kyungpook National University, Daegu, 41566, Republic of Korea

    Junhyung Ko & Young Jin Suh

Authors
  1. Juan de Dios Pérez
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Imsoon Jeong
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Junhyung Ko
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Young Jin Suh
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Young Jin Suh.

Additional information

This work was supported by Grant Project no. NRF-2015-R1A2A1A-01002459 from National Research Foundation of Korea. J. de Dios Pérez was supported by MCT-FEDER Project MTM-2013-47828-C2-1-P, I. Jeong, and J. Ko were supported by NRF-2017-R1A2B4005317, and Y. J. Suh by Bokhyun Research 2017.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

de Dios Pérez, J., Jeong, I., Ko, J. et al. Real Hypersurfaces with Killing Shape Operator in the Complex Quadric . Mediterr. J. Math. 15, 6 (2018). https://doi.org/10.1007/s00009-017-1052-1

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s00009-017-1052-1

Mathematics Subject Classification

Keywords

Search

Navigation