Advertisement

Israel Journal of Mathematics

, Volume 202, Issue 1, pp 255–274 | Cite as

On the geometry of flat pseudo-Riemannian homogeneous spaces

  • Wolfgang Globke
Article

Abstract

Let M = ℝ s n /Γ be a complete flat pseudo-Riemannian homogeneous manifold, Γ ⊂ Iso(ℝ s n ) its fundamental group and G the Zariski closure of Γ in Iso(ℝ s n ). We show that the G-orbits in ℝ s n are affine subspaces and affinely diffeomorphic to G endowed with the (0)-connection. If the restriction of the pseudo-scalar product on ℝ s n to the G-orbits is nondegenerate, then M has abelian linear holonomy. If additionally G is not abelian, then G contains a certain subgroup of dimension 6. In particular, for non-abelian G, orbits with non-degenerate metric can appear only if dim G ≥ 6. Moreover, we show that ℝ s n is a trivial algebraic principal bundle GM → ℝ n−k . As a consquence, M is a trivial smooth bundle G/Γ → M → ℝ n−k with compact fiber G/Γ.

Keywords

Homogeneous Space Algebraic Group Unipotent Group Zariski Closure Algebraic Subgroup 
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.

Preview

Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.

References

  1. [1]
    O. Baues, Prehomogeneous affine representations and flat pseudo-Riemannian manifolds, in Handbook of Pseudo-Riemannian Geometry and Supersymmetry, IRMA Lectures in Mathematics and Theoretical Physics, Vol. 16, European Mathematical Society, Zürich, 2010, pp. 731–817 (also arXiv:0809.0824v1).CrossRefGoogle Scholar
  2. [2]
    O. Baues and W. Globke, Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group, Proceedings of the American Mathematical Society 140 (2012), 2479–2488 (also arXiv:1009.3383).MathSciNetCrossRefMATHGoogle Scholar
  3. [3]
    A. Borel, Linear Algebraic Groups, 2nd edition, Springer, Berlin, 1991.CrossRefMATHGoogle Scholar
  4. [4]
    L. Corwin and F. P. Greenleaf, Representations of Nilpotent Lie Groups and their Applications, Cambrideg, Studies in Advanced Mathematics, Vol. 18, Cambridge University Press, 1990.Google Scholar
  5. [5]
    W. Globke, Holonomy Groups of Flat Pseudo-Riemannian Homogeneous Manifolds, Dissertation, Karlsruhe Institute of Technology, 2011.Google Scholar
  6. [6]
    W. Globke, Holonomy groups of complete flat pseudo-Riemannian homogeneous spaces, Advances in Mathematics 240 (2013), 88–105 (also arXiv:1205.3285).MathSciNetCrossRefMATHGoogle Scholar
  7. [7]
    H. Kraft and G. W. Schwarz, Reductive group actions with one-dimensional quotient, Publications Mathématiques. Institut de Hautes Études Scientifiques 76 (1992), 1–97.MathSciNetCrossRefMATHGoogle Scholar
  8. [8]
    B. O’Neill, Semi-Riemannian Geometry, Pure and Applied Mathematics, Vol. 103, Academic Press, New York, 1983.MATHGoogle Scholar
  9. [9]
    A. Onishchik and É. Vinberg, eds., Lie Groups and Lie Algebras III, Encyclopedia of Mathematical Sciences, Vol. 41, Springer, Berlin, 1994.MATHGoogle Scholar
  10. [10]
    D. Perrin, Algebraic Geometry, Springer, London, 2008.CrossRefMATHGoogle Scholar
  11. [11]
    M. S. Raghunathan, Discrete Subgroups of Lie Groups, Ergebnisse der Mathematik und ihrer Grenzgebiete, Vol. 68, Springer, New York-Heidelberg, 1972.CrossRefMATHGoogle Scholar
  12. [12]
    M. Rosenlicht, Some basic theorems on algebraic groups, American Journal of Mathematics 78 (1956), 401–443.MathSciNetCrossRefMATHGoogle Scholar
  13. [13]
    M. Rosenlicht, Questions of rationality for solvable algebraic groups over nonperfect fields, Annali di Matematica Pura ed Applicata 62 (1963), 97–120.MathSciNetCrossRefGoogle Scholar
  14. [14]
    N. Steenrod, The Topology of Fibre Bundles, Princeton Mathematical Series, Vol. 14, Princeton University Press, Priceton, NJ, 1951.MATHGoogle Scholar
  15. [15]
    J. A. Wolf, Homogeneous manifolds of zero curvature, Transactions of the American Mathematical Society 104 (1962), 462–469.MathSciNetCrossRefGoogle Scholar
  16. [16]
    J. A. Wolf, Spaces of Constant Curvature, 6th edition, American Mathematical Society, Providence, RI, 2011.MATHGoogle Scholar

Copyright information

© Hebrew University Magnes Press 2014

Authors and Affiliations

  1. 1.School of Mathematical SciencesThe University of AdelaideAdelaideAustralia

Personalised recommendations