Skip to main content
SpringerLink
Log in

The Solvable Models of Noncompact Real Two-Plane Grassmannians and Some Applications

  • Conference paper
  • First Online:
Hermitian–Grassmannian Submanifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 203))

  • 613 Accesses

Abstract

Every Riemannian symmetric space of noncompact type is isometric to some solvable Lie group equipped with a left-invariant Riemannian metric. The corresponding metric solvable Lie algebra is called the solvable model of the symmetric space. In this paper, we give explicit descriptions of the solvable models of noncompact real two-plane Grassmannians, and mention some applications to submanifold geometry, contact geometry, and geometry of left-invariant metrics.

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

Access this chapter

Log in via an institution
Chapter
USD 29.95
Price excludes VAT (USA)
  • Available as PDF
  • Read on any device
  • Instant download
  • Own it forever
eBook
USD 129.00
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 169.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 169.99
Price excludes VAT (USA)
  • Durable hardcover edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info

Tax calculation will be finalised at checkout

Purchases are for personal use only

Institutional subscriptions

References

  1. Berndt, J.: Homogeneous hypersurfaces in hyperbolic spaces. Math. Z. 229, 589–600 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  2. Berndt, J., Suh, Y.J.: Contact hypersurfaces in Kähler manifolds. Proc. Am. Math. Soc. 143, 2637–2649 (2015)

    Article  MATH  Google Scholar 

  3. Berndt, J., Tamaru, H.: Homogeneous codimension one foliations on noncompact symmetric spaces. J. Differ. Geom. 63, 1–40 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  4. Berndt, J., Tricerri, F., Vanhecke, L.: Generalized Heisenberg Groups and Damek-Ricci Harmonic Spaces. Lecture Notes in Mathematics, vol. 1598. Springer, Berlin (1995)

    Google Scholar 

  5. Berndt, J., Díaz-Ramos, J.C., Tamaru, H.: Hyperpolar homogeneous foliations on symmetric spaces of noncompact type. J. Differ. Geom. 86, 191–235 (2010)

    Article  MathSciNet  MATH  Google Scholar 

  6. Blair, D.E., Koufogiorgos, T., Papantoniou, B.J.: Contact metric manifolds satisfying a nullity condition. Israel J. Math. 91, 189–214 (1995)

    Article  MathSciNet  MATH  Google Scholar 

  7. Boeckx, E.: A full classification of contact metric \((k,\mu )\)-spaces. Illinois J. Math. 44, 212–219 (2000)

    Google Scholar 

  8. Cho, J.T., Hashinaga, T., Kubo, A., Taketomi, Y., Tamaru, H.: Realizations of some contact metric manifolds as Ricci soliton real hypersurfaces. Preprint, arXiv:1702.07256

  9. Ghosh, A., Sharma, R.: A classification of Ricci solitons as \((k, \mu )\)-contact metrics. In: Suh, Y.J., Berndt, J., Ohnita, Y., Kim, B.H., Lee, H. (eds.) Real and Complex Submanifolds. Springer Proceedings in Mathematics and Statistics, vol. 106, pp. 349–358. Springer, Tokyo (2014)

    Google Scholar 

  10. Hashinaga, T., Kajigaya, T.: A class of non-compact homogeneous Lagrangian submanifolds in complex hyperbolic spaces. Ann. Global Anal. Geom. 51, 21–33 (2017)

    Google Scholar 

  11. Hashinaga, T., Kubo, A., Tamaru, H.: Some topics of homogeneous submanifolds in complex hyperbolic spaces. In: Maeda, S., Ohnita, Y., Cheng, Q.-M. (eds.) Differential Geometry of Submanifolds and its Related Topics, pp. 230–244. World Scientific (2013)

    Google Scholar 

  12. Heber, J.: Noncompact homogeneous Einstein spaces. Invent. Math. 133, 279–352 (1998)

    Article  MathSciNet  MATH  Google Scholar 

  13. Kodama, H., Takahara, A., Tamaru, H.: The space of left-invariant metrics on a Lie group up to isometry and scaling. Manuscripta Math. 135, 229–243 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  14. Kubo, A.: Geometry of homogeneous polar foliations of complex hyperbolic spaces. Hiroshima Math. J. 45, 109–123 (2015)

    MathSciNet  MATH  Google Scholar 

  15. Kubo, A., Tamaru, H.: A sufficient condition for congruency of orbits of Lie groups and some applications. Geom. Dedicata 167, 233–238 (2013)

    Article  MathSciNet  MATH  Google Scholar 

  16. Lauret, J.: Degenerations of Lie algebras and geometry of Lie groups. Differ. Geom. Appl. 18, 177–194 (2003)

    Article  MathSciNet  MATH  Google Scholar 

  17. Lauret, J.: Ricci soliton solvmanifolds. J. reine angew. Math. 650, 1–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  18. Milnor, J.: Curvatures of left invariant metrics on Lie groups. Adv. Math. 21, 293–329 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  19. Ryan, P.: Intrinsic properties of real hypersurfaces in complex space forms. In: Chen, W.H., Wang, C.P., Li, A.-M., Simon, U., Wiehe, M., Verstraelen, L. (eds.) Geometry and Topology of Submanifolds, vol. X, pp. 266–273. World Scientific (2000)

    Google Scholar 

Download references

Acknowledgements

The first author was supported in part by Basic Science Research Program through the National Research Foundation of Korea (NRF) funded by the Ministry of Education, Science and Technology (2014R1A1A2053665). The second author was supported in part by JSPS KAKENHI (16K17603). The fifth author was supported in part by JSPS KAKENHI (26287012, 16K13757).

Author information

Authors and Affiliations

  1. Department of Mathematics, Chonnam National University, Gwangju, 61186, Korea

    Jong Taek Cho

  2. National Institute of Technology, Kitakyushu College, Kitakyushu, 802-0985, Japan

    Takahiro Hashinaga

  3. Faculty of Economic Sciences, Hiroshima Shudo University, Hiroshima, 731-3195, Japan

    Akira Kubo

  4. Department of Mathematics, Hiroshima University, Higashi-Hiroshima, 739-8526, Japan

    Yuichiro Taketomi & Hiroshi Tamaru

Authors
  1. Jong Taek Cho
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Takahiro Hashinaga
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Akira Kubo
    View author publications

    You can also search for this author in PubMed Google Scholar

  4. Yuichiro Taketomi
    View author publications

    You can also search for this author in PubMed Google Scholar

  5. Hiroshi Tamaru
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Hiroshi Tamaru .

Editor information

Editors and Affiliations

  1. Department of Mathematics and RIRCM, Kyungpook National University , Daegu, Korea (Republic of)

    Young Jin Suh

  2. Department of Mathematics and OCAMI, Osaka City University, Osaka, Japan

    Yoshihiro Ohnita

  3. School of Mathematics and Statistics, Southwest University, Chongqing, China

    Jiazu Zhou

  4. Department of Applied Mathematics, Kyung Hee University, Gyeonggi, Korea (Republic of)

    Byung Hak Kim

  5. Research Institute of Real and Complex Manifolds (RIRCM), Kyungpook National University, Daegu, Korea (Republic of)

    Hyunjin Lee

Rights and permissions

Reprints and permissions

Copyright information

© 2017 Springer Nature Singapore Pte Ltd.

About this paper

Cite this paper

Cho, J.T., Hashinaga, T., Kubo, A., Taketomi, Y., Tamaru, H. (2017). The Solvable Models of Noncompact Real Two-Plane Grassmannians and Some Applications. In: Suh, Y., Ohnita, Y., Zhou, J., Kim, B., Lee, H. (eds) Hermitian–Grassmannian Submanifolds. Springer Proceedings in Mathematics & Statistics, vol 203. Springer, Singapore. https://doi.org/10.1007/978-981-10-5556-0_26

Download citation

Publish with us

Policies and ethics

Search

Navigation