Skip to main content
SpringerLink
Log in

The Riemannian manifolds with boundary and large symmetry

  • Published:
Chinese Annals of Mathematics, Series B Aims and scope Submit manuscript

Abstract

Seventy years ago, Myers and Steenrod showed that the isometry group of a Riemannian manifold without boundary has a structure of Lie group. In 2007, Bagaev and Zhukova proved the same result for a Riemannian orbifold. In this paper, the authors first show that the isometry group of a Riemannian manifold M with boundary has dimension at most ½ dimM(dimM − 1). Then such Riemannian manifolds with boundary that their isometry groups attain the preceding maximal dimension are completely classified.

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. Bagaev, A. V. and Zhukova, N. I., The isometry groups of Riemannian orbifolds, Sib. Math. J., 48, 2007, 579–592.

    Article  MathSciNet  Google Scholar 

  2. Bredon, G. E., Introduction to Compact Transformation Groups, Pure and Applied Mathematics, Vol. 46, Academic Press, New York, 1972.

    Google Scholar 

  3. van Dantzig, D. and van der Waerden, B. L., Über metrisch homogene Räume, Abh. Math. Sem. Univ. Hamburg, 6, 1928, 374–376.

    Article  Google Scholar 

  4. Hatcher, A., Algebraic Topology, Cambridge University Press, Cambridge, 2002.

    MATH  Google Scholar 

  5. Kawakubo, K., The Theory of Transformation Groups, Oxford University Press, New York, 1991.

    MATH  Google Scholar 

  6. Kobayashi, S., Transformation Groups in Differential Geometry, Ergebnisse der Mathematik und Ihrer Grenzgebiete, Vol. 70. Springer-Verlag, Heidelberg, 1972.

    Google Scholar 

  7. Kobayshi, S. and Nomizu, K., Foundations of Differential Geometry, Vol. I, Interscience Publishers, New York, 1963.

    Google Scholar 

  8. Milnor, J. W. and Stasheff, J. D., Characteristic Classes, Princeton University Press, Princeton, 1974.

    MATH  Google Scholar 

  9. Myers, S. B. and Steenrod, N. E., The group of isometries of a Riemannian manifold, Ann. of Math. (2), 40, 1939, 400–416.

    Article  MathSciNet  Google Scholar 

  10. Petersen, P., Riemannian Geometry, 2nd ed., Graduate Texts in Mathematics, Vol. 171, Springer-Verlag, New York, 2006.

    Google Scholar 

Download references

Author information

Authors and Affiliations

  1. Department of Mathematics, University of Science and Technology of China, Hefei, 230026, China

    Zhi Chen, Yiqian Shi & Bin Xu

Authors
  1. Zhi Chen
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Yiqian Shi
    View author publications

    You can also search for this author in PubMed Google Scholar

  3. Bin Xu
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Bin Xu.

Additional information

Project supported by the National Natural Science Foundation of China (Nos. 10601053, 10671096, 10871184, 10971104).

Rights and permissions

Reprints and permissions

About this article

Cite this article

Chen, Z., Shi, Y. & Xu, B. The Riemannian manifolds with boundary and large symmetry. Chin. Ann. Math. Ser. B 31, 347–360 (2010). https://doi.org/10.1007/s11401-009-0037-1

Download citation

  • Received:

  • Revised:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11401-009-0037-1

Keywords

2000 MR Subject Classification

Search

Navigation