Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Flat pseudo-Riemannian homogeneous spaces with non-abelian holonomy group


Authors: Oliver Baues and Wolfgang Globke
Journal: Proc. Amer. Math. Soc. 140 (2012), 2479-2488
MSC (2010): Primary 53C30, 57S30; Secondary 20G05
DOI: https://doi.org/10.1090/S0002-9939-2011-11080-3
Published electronically: October 27, 2011
MathSciNet review: 2898710
Full-text PDF Free Access

Abstract | References | Similar Articles | Additional Information

Abstract: We construct homogeneous flat pseudo-Riemannian manifolds
with non-abelian fundamental group. In the compact case, all homogeneous flat pseudo-Riemannian manifolds are complete and have abelian linear holonomy group. To the contrary, we show that there do exist non-compact and non-complete examples, where the linear holonomy is non-abelian, starting in dimensions $ \geq 8$, which is the lowest possible dimension. We also construct a complete flat pseudo-Riemannian homogeneous manifold of dimension 14 with non-abelian linear holonomy. Furthermore, we derive a criterion for the properness of the action of an affine transformation group with transitive centralizer.


References [Enhancements On Off] (What's this?)

  • 1. O. Baues, Flat Pseudo-Riemannian manifolds and prehomogeneous affine representations, in `Handbook of Pseudo-Riemannian Geometry and Supersymmetry', EMS, IRMA Lect. Math. Theor. Phys. 16, 2010, pp. 731-817. MR 2681607
  • 2. D. Duncan, E. Ihrig, Incomplete flat homogeneous geometries, Differential geometry: Geometry in mathematical physics and related topics (Los Angeles, CA, 1990), 197-202, Proc. Sympos. Pure Math., 54, Part 2, Amer. Math. Soc., Providence, RI, 1993. MR 1216539 (94f:53115)
  • 3. D. Duncan, E. Ihrig, Flat pseudo-Riemannian manifolds with a nilpotent transitive group of isometries, Ann.Global Anal. Geom. 10 (1992), no. 1, 87-101. MR 1172623 (93f:53055)
  • 4. D. Duncan, E. Ihrig, Homogeneous spacetimes of zero curvature, Proc. Amer. Math. Soc. 107 (1989), no. 3, 785-795. MR 975639 (90b:53075)
  • 5. J. Marsden, On completeness of homogeneous pseudo-riemannian manifolds, Indiana Univ. Math. J. 22 (1972/73), 1065-1066. MR 0319128 (47:7674)
  • 6. A. Püttmann, Free affine actions of unipotent groups on $ \mathbb{C}^n$, Transform. Groups 12 (2007), no. 1, 137-151. MR 2308033 (2008g:14121)
  • 7. J.A. Wolf, Spaces of constant curvature, 6th edition, Amer. Math. Soc., 2011. MR 2742530
  • 8. J.A. Wolf, Homogeneous manifolds of zero curvature, Trans. Amer. Math. Soc. 104 (1962), 462-469. MR 0140050 (25:3474)
  • 9. J.A. Wolf, Flat homogeneous pseudo-Riemannian manifolds, Geom. Dedicata 57 (1995), no. 1, 111-120. MR 1344776 (96f:53070)
  • 10. J.A. Wolf, Isoclinic spheres and flat homogeneous pseudo-Riemannian manifolds, Crystallographic groups and their generalizations (Kortrijk, 1999), 303-310, Contemp. Math., 262, Amer. Math. Soc., Providence, RI, 2000. MR 1796139 (2002c:53118)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2010): 53C30, 57S30, 20G05

Retrieve articles in all journals with MSC (2010): 53C30, 57S30, 20G05


Additional Information

Oliver Baues
Affiliation: Department of Mathematics, Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Email: baues@kit.edu

Wolfgang Globke
Affiliation: Department of Mathematics, Institute for Algebra and Geometry, Karlsruhe Institute of Technology, 76128 Karlsruhe, Germany
Email: globke@math.uni-karlsruhe.de

DOI: https://doi.org/10.1090/S0002-9939-2011-11080-3
Received by editor(s): October 1, 2010
Received by editor(s) in revised form: February 16, 2011
Published electronically: October 27, 2011
Communicated by: Jianguo Cao
Article copyright: © Copyright 2011 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.

American Mathematical Society