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A Flat Nonmetrizable Connexion

Published online by Cambridge University Press:  20 November 2018

H. G. Helfenstein
H. G. Helfenstein*
Affiliation:
University of Ottawa, Ottawa, Ontario
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A linear connexion ∇ on a smooth manifold M with vanishing torsion and curvature is called flat. It is well known that such a connexion need not be the Levi-Civita connexion of a Riemannian (nor pseudo-Riemannian) structure (cf. [1]).

The following example with M homeomorphic to a cylinder is particularly simple. Using polar coordinates the punctured plane M=R2\{0} is covered by a smooth atlas with two charts

Type
Research Article
Information
Canadian Mathematical Bulletin , Volume 15 , Issue 2 , June 1972 , pp. 205 - 206
Copyright
Copyright © Canadian Mathematical Society 1972

References

1. Kobayashi, S. and Nomizu, K., Foundations of differential geometry, Vol. I, Interscience, New York, 1963.Google Scholar