Skip to main content
SpringerLink
Log in

The Space of Left-Invariant Riemannian Metrics

  • Conference paper
  • First Online:
Geometry and Topology of Manifolds

Part of the book series: Springer Proceedings in Mathematics & Statistics ((PROMS,volume 154))

  • 1716 Accesses

Abstract

Geometry of left-invariant Riemannian metrics on Lie groups has been studied very actively. We have proposed a new framework for studying this topic from the viewpoint of the space of left-invariant metrics. In this expository paper, we introduce our framework, and mention two results. One is a generalization of Milnor frames, and another is a characterization of solvsolitons of dimension three in terms of submanifold geometry.

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 84.99
Price excludes VAT (USA)
  • Available as EPUB and PDF
  • Read on any device
  • Instant download
  • Own it forever
Softcover Book
USD 109.99
Price excludes VAT (USA)
  • Compact, lightweight edition
  • Dispatched in 3 to 5 business days
  • Free shipping worldwide - see info
Hardcover Book
USD 109.99
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. Alekseevskii, D., Kimel’fel’d, B.: Structure of homogeneous Riemannian spaces with zero Ricci curvature. Funct. Anal. Appl. 9, 97–102 (1975)

    Article  MathSciNet  MATH  Google Scholar 

  2. Azencott, R., Wilson, E.: Homogeneous manifolds with negative curvature I. Trans. Am. Math. Soc. 215, 323–362 (1976)

    Article  MathSciNet  MATH  Google Scholar 

  3. Azencott, R., Wilson, E.: Homogeneous manifolds with negative curvature II. Mem. Am. Math. Soc. 8 (1976)

    Google Scholar 

  4. Berndt, J., Brück, M.: Cohomogeneity one actions on hyperbolic spaces. J. Reine Angew. Math. 541, 209–235 (2001)

    Article  MathSciNet  MATH  Google Scholar 

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

    MathSciNet  MATH  Google Scholar 

  6. Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces with a totally geodesic singular orbit. Tohoku Math. J. 56, 163–177 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  7. Berndt, J., Tamaru, H.: Cohomogeneity one actions on noncompact symmetric spaces of rank one. Trans. Am. Math. Soc. 359, 3425–3438 (2007)

    Article  MathSciNet  MATH  Google Scholar 

  8. Berndt, J., Tamaru, H.: Cohomogeneity one actions on symmetric spaces of noncompact type. J. Reine Angew. Math. 683, 129–159 (2013)

    MathSciNet  MATH  Google Scholar 

  9. Besse, A.: Einstein manifolds. Classics in Mathematics. Springer, Berlin (2008)

    Google Scholar 

  10. Böhm, C., Wang, M.Y., Ziller, W.: A variational approach for compact homogeneous Einstein manifolds. Geom. Funct. Anal. 14, 681–733 (2004)

    Article  MathSciNet  MATH  Google Scholar 

  11. Chebarikov, M.S.: On the Ricci curvature of three-dimensional metric Lie algebras (Russian). Vladikavkaz. Mat. Zh. 16, 57–67 (2014)

    MATH  Google Scholar 

  12. Chow, B., Knopf, D.: The Ricci flow: an introduction. In: Mathematical Surveys and Monographs, vol. 110. American Mathematical Society, Providence, RI (2004)

    Google Scholar 

  13. D’Atri, J.E., Ziller, W.: Naturally reductive metrics and Einstein metrics on compact Lie groups. Mem. Amer. Math. Soc. 18 (1979)

    Google Scholar 

  14. Ha, K.Y., Lee, J.B.: Left invariant metrics and curvatures on simply connected three-dimensional Lie groups. Math. Nachr. 282, 868–898 (2009)

    Article  MathSciNet  MATH  Google Scholar 

  15. Hashinaga, T.: On the minimality of the corresponding submanifolds to four-dimensional solvsolitons. Hiroshima Math. J. 44, 173–191 (2014)

    MathSciNet  MATH  Google Scholar 

  16. Hashinaga, T., Tamaru, H.: Three-dimensional solvsolitons and the minimality of the corresponding submanifolds. Preprint, arXiv:1501.05513

  17. Hashinaga, T., Tamaru, H., Terada, K.: Milnor-type theorems for left-invariant Riemannian metrics on Lie groups. J. Math. Soc. Jpn., to appear

    Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  19. Heintze, E.: On homogeneous manifolds of negative curvature. Math. Ann. 211, 23–34 (1974)

    Article  MathSciNet  MATH  Google Scholar 

  20. Jablonski, M.: Homogeneous Ricci solitons. J. Reine Angew. Math. 699, 159–182 (2015)

    Google Scholar 

  21. Kerr, M.: New examples of non-symmetric Einstein solvmanifolds of negative Ricci curvature. Ann. Global Anal. Geom. 46, 281–291 (2014)

    Article  MathSciNet  MATH  Google Scholar 

  22. 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 

  23. Kremlev, A.G., Nikonorov, Yu.G.: The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The unimodular case. Siberian Adv. Math. 19, 245–267 (2009)

    Google Scholar 

  24. Kremlev, A.G., Nikonorov, Yu.G.: The signature of the Ricci curvature of left-invariant Riemannian metrics on four-dimensional Lie groups. The nonunimodular case. Siberian Adv. Math. 20, 1–57 (2010)

    Google Scholar 

  25. Kubo, A., Onda, K., Taketomi, Y., Tamaru, H.: On the moduli spaces of left-invariant pseudo-Riemannian metrics on Lie groups. Preprint, arXiv:1509.08336

  26. Lafuente, R., Lauret, J.: Structure of homogeneous Ricci solitons and the Alekseevskii conjecture. J. Differ. Geom. 98, 315–347 (2014)

    MathSciNet  MATH  Google Scholar 

  27. Lauret, J.: Ricci soliton homogeneous nilmanifolds. Math. Ann. 319, 715–733 (2001)

    Article  MathSciNet  MATH  Google Scholar 

  28. Lauret, J.: Einstein solvmanifolds and nilsolitons. In: New developments in Lie theory and geometry. Contemp. Math. 491, 1–35. Amer. Math. Soc., Providence, RI (2009)

    Google Scholar 

  29. Lauret, J.: Ricci soliton solvmanifolds. J. Reine Angew. Math. 650, 1–21 (2011)

    Article  MathSciNet  MATH  Google Scholar 

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

    Article  MathSciNet  MATH  Google Scholar 

  31. Mori, K.: Einstein metrics on Boggino-Damek-Ricci-type solvable Lie groups. Osaka J. Math. 39, 345–362 (2002)

    MathSciNet  MATH  Google Scholar 

  32. Tamaru, H.: A class of noncompact homogeneous Einstein manifolds. In: Differential Geometry and Its Applications, pp. 119–127. Matfyzpress, Prague (2005)

    Google Scholar 

  33. Tamaru, H.: Noncompact homogeneous Einstein manifolds attached to graded Lie algebras. Math. Z. 259, 171–186 (2008)

    Article  MathSciNet  MATH  Google Scholar 

  34. Tamaru, H.: Parabolic subgroups of semisimple Lie groups and Einstein solvmanifolds. Math. Ann. 351, 51–66 (2011)

    Article  MathSciNet  MATH  Google Scholar 

  35. Taketomi, Y., Tamaru, H.: On the nonexistence of left-invariant Ricci solitons—a conjecture and examples. Preprint

    Google Scholar 

  36. Wallach, N.: Compact homogeneous Riemannian manifolds with strictly positive curvature. Ann. Math. 96, 277–295 (1972)

    Google Scholar 

Download references

Acknowledgments

The author is grateful to the referee for careful reading of the manuscript. This work was supported by JSPS KAKENHI Grant Number 24654012.

Author information

Authors and Affiliations

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

    Hiroshi Tamaru

Authors
  1. 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. The University of Tokyo, Meguro-ku, Tokyo, Japan

    Akito Futaki

  2. Mathematical Institute, Tohoku University, Sendai,Aoba-ku, Japan

    Reiko Miyaoka

  3. School of Mathematical Sciences, Beijing Normal University, Beijing, China

    Zizhou Tang

  4. Chern Institute of Mathematics, Nankai University, Tianjin, China

    Weiping Zhang

Rights and permissions

Reprints and permissions

Copyright information

© 2016 Springer Japan

About this paper

Cite this paper

Tamaru, H. (2016). The Space of Left-Invariant Riemannian Metrics. In: Futaki, A., Miyaoka, R., Tang, Z., Zhang, W. (eds) Geometry and Topology of Manifolds. Springer Proceedings in Mathematics & Statistics, vol 154. Springer, Tokyo. https://doi.org/10.1007/978-4-431-56021-0_17

Download citation

Publish with us

Policies and ethics

Search

Navigation