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)



Locally homogeneous affine connections on compact surfaces

Author: Barbara Opozda
Journal: Proc. Amer. Math. Soc. 132 (2004), 2713-2721
MSC (2000): Primary 53C05, 53C40
Published electronically: April 9, 2004
MathSciNet review: 2054798
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: Global properties of locally homogeneous and curvature homogeneous affine connections are studied. It is proved that the only locally homogeneous connections on surfaces of genus different from 1 are metric connections of constant curvature. There exist nonmetrizable nonlocally symmetric locally homogeneous affine connections on the torus of genus 1. It is proved that there is no global affine immersion of the torus endowed with a nonflat locally homogeneous connection into ${\mathbf R} ^3$.

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

  • 1. G. S. Birman and K. Nomizu, The Gauss-Bonnet theorem for $2$-dimensional spacetimes, Mich. Math. J., 31 (1984) 77-81. MR 85g:53073
  • 2. C. Godbillon, Dynamical Systems on Surfaces, Springer-Verlag, Berlin, Heidelberg, New York, 1983. MR 84b:57018
  • 3. J. Milnor, On the existence of a connection of curvature zero, Comment. Math. Helv., 32 (1958) 215-223. MR 20:2020
  • 4. S. Kobayashi and K. Nomizu, Foundations of Differential Geometry, vol. II, Interscience, Wiley, New York, 1969. MR 38:6501
  • 5. O. Kowalski, B. Opozda, and Z. Vlásek, A classification of locally homogeneous affine connections with skew-symmetric Ricci tensor on $2$-dimensional manifolds, Monatshefte für Mathematik, 130 (2000), 109-125. MR 2001f:53048
  • 6. K. Nomizu and T. Sasaki, Affine differential geometry, Cambridge University Press, Cambridge, 1994. MR 96e:53014
  • 7. B. Opozda, Locally symmetric connections on surfaces, Results in Math., 20 (1991), 725-743. MR 93b:53014
  • 8. B. Opozda, Curvature homogeneous and locally homogeneous affine connections, Proc. Amer. Math. Soc., 124 (1996), 1889-1893. MR 96h:53030
  • 9. B. Opozda, Affine versions of Singer's theorem on locally homogeneous spaces, Ann. Global Anal. Geom., 15 (1997), 187-199. MR 98d:53032
  • 10. B. Opozda, On locally homogeneous $G$-structures, Geom. Dedicata, 73 (1998), 215-223. MR 99k:53042
  • 11. B. Opozda, A new cylinder theorem, Math. Ann., 312 (1998), 1-12. MR 2000j:53074
  • 12. B. Opozda, A classification of locally homogeneous connections on $2$-dimensional manifolds, to appear in J. Diff. Geom. Appl.

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC (2000): 53C05, 53C40

Retrieve articles in all journals with MSC (2000): 53C05, 53C40

Additional Information

Barbara Opozda
Affiliation: Instytut Matematyki Uniwersytet Jagielloński, ul. Reymonta 4, 30-059 Kraków, Poland

Keywords: Affine connection, local homogeneity
Received by editor(s): March 3, 2003
Received by editor(s) in revised form: June 16, 2003
Published electronically: April 9, 2004
Communicated by: Jon G. Wolfson
Article copyright: © Copyright 2004 American Mathematical Society

American Mathematical Society