Hostname: page-component-78c5997874-t5tsf Total loading time: 0 Render date: 2024-11-15T10:33:50.571Z Has data issue: false hasContentIssue false
  • English
  • Français

Parallel Translation in Vector Bundles with Abelian Structure Group and the Gauss-Bonnet Formula

Published online by Cambridge University Press:  20 November 2018

Hansklaus Rummler
Hansklaus Rummler*
Affiliation:
Universite de Fribourg, Fribourg, Switzerland
Rights & Permissions [Opens in a new window]

Extract

Core share and HTML view are not available for this content. However, as you have access to this content, a full PDF is available via the ‘Save PDF’ action button.

Most proofs for the classical Gauss-Bonnet formula use special coordinates, or other non-trivial preparations. Here, a simple proof is given, based on the fact that the structure group SO(2) of the tangent bundle of an oriented 2-dimensional Riemannian manifold is abelian. Since only this hypothesis is used, we prove a slightly more general result (Theorem 1).

Type
Research Article
Information
Canadian Journal of Mathematics , Volume 25 , Issue 4 , August 1973 , pp. 765 - 771
Copyright
Copyright © Canadian Mathematical Society 1973

References

1. Greub, W. H., Eine geometrische Deutung des Krümmungstensors, Arch. Math. 2 (1949), 148–151.Google Scholar
2. Greub, W. H., Zur Theorie der linearen Uebertragungen, Ann. Acad. Sci. Fenn., Series A I. 346 (1964), 1–22.Google Scholar
3. Greub, W. H., Halperin, S., and Vanstone, J. R., Curvature, connections and cohomology, Vol. I (Academic Press, New York, 1972).Google Scholar
4. Hirzebruch, F., Topological methods in algebraic geometry (Springer, Berlin-Heidelberg-New York, 1966).Google Scholar