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manuscripta mathematica

, Volume 93, Issue 1, pp 205–217 | Cite as

On space forms of Grassmann manifolds

  • Brett McInnes
Article
  • 25 Downloads

Abstract

We show that, in each odd dimensionn =m 2, there is a class of Grassmann quotient spaces not included in Wolf’s classic solution of the Grassmann space form problem. We classify all of these new Grassmann space forms up to isometry. As an application, we exhibit a pair of compact Einstein manifolds of dimensionm 2 with holonomy groups which are abstractly isomorphic yet not conjugate in the orthogonal group, thus proving that a theorem of Besse cannot be extended to non-simply-connected Einstein manifolds.

Keywords

Riemannian Manifold Symmetric Space Space Form Maximal Torus Einstein Manifold 
These keywords were added by machine and not by the authors. This process is experimental and the keywords may be updated as the learning algorithm improves.

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References

  1. [1]
    Besse, A.: Einstein manifolds. Berlin, Heidelberg. New York: Springer 1987.zbMATHGoogle Scholar
  2. [2]
    Kobayashi, S. and Nomizu, K.: Foundations of differential geometry, Vol. II. New York, London, Sydney: Interscience 1969.zbMATHGoogle Scholar
  3. [3]
    McInnes, B.: Holonomy groups of compact Riemannian manifolds: a classification in dimensions up to ten. J. Math. Phys. 34, 4273–4286 (1993).zbMATHCrossRefGoogle Scholar
  4. [4]
    McInnes, B.: Examples of Einstein manifolds with all possible holonomy groups in dimensions less than seven. J. Math. Phys. 34, 4287–4304 (1993).zbMATHCrossRefGoogle Scholar
  5. [5]
    O’Neill, B.: Semi-riemannian geometry. New York: Academic Press 1983.zbMATHGoogle Scholar
  6. [6]
    de Siebenthal, J.: Sur les groupes de Lie compacts non connexes. Commentarii Mathematici Helvetici 31, 41–89 (1956).zbMATHCrossRefGoogle Scholar
  7. [7]
    Wolf, J. A.: Spaces of constant curvature. Wilmington: Publish or Perish 1984.Google Scholar
  8. [8]
    Wolf, J. A.: Discrete groups, symmetric spaces, and global holonomy. Amer. J. Math. 84, 527–542 (1962).zbMATHCrossRefGoogle Scholar

Copyright information

© Springer-Verlag 1997

Authors and Affiliations

  • Brett McInnes
    • 1
  1. 1.Department of MathematicsNational University of SingaporeSingaporeRepublic of Singapore

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