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On Affine Transformations of a Riemannian Manifold*

Published online by Cambridge University Press:  22 January 2016

Jun-Ichi Hano
Jun-Ichi Hano*
Affiliation:
Mathematical Institute, Nagoya University
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In this paper we establish some theorems about the group of affine transformations on a Riemannian manifold. First we prove a decomposition theorem (Theorem 1) of the largest connected group of affine transformations on a simply connected complete Riemannian manifold, which corresponds to the decomposition theorem of de Rham [4] for the manifold. In the case of the largest group of isometries, a theorem of the same type is found in de Rham’s paper [4] in a weaker form. Using Theorem 1 we obtain a sufficient condition for an infinitesimal affine transformation to be a Killing vector field (Theorem 2). This result includes K. Yano’s theorem [13] which states that on a compact Riemannian manifold an infinitesimal affine transformation is always a Killing vector field. His proof of the theorem depends on an integral formula which is valid only for a compact manifold. Our method is quite different and is based on a result [11] of K. Nomizu.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 99 - 109
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

Footnotes

*

The subject of this paper was prepared while the author was a Yukawa Fellow at Osaka University.

References

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