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A Theorem on the Affine Transformation Group of a Riemannian Manifold

Published online by Cambridge University Press:  22 January 2016

Shoshichi Kobayashi
Shoshichi Kobayashi*
Affiliation:
University of Washington
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Every Riemannian manifold has a unique affine connection without torsion, which is necessarily invariant by any isometrical transformation of the manifold. However, an affine transformation (i.e., transformation leaving invariant the affine connection) is not necessarily an isometrical transformation. (Consider, for example, the ordinary Euclidean space).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 39 - 41
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

[1] Nomizu, K.: Sur les transformations affines d’une variété riemanniennes. C. R. Acad. Paris, 237 (1953), 1308–1310. Also, Studies on Riemannian homogeneous spaces (in this journal).Google Scholar
[2] Yano, K.: On harmonic and Killing vector fields. Ann. Math. 55 (1952), 38–45.Google Scholar