Hostname: page-component-cd9895bd7-lnqnp Total loading time: 0 Render date: 2024-12-23T23:45:42.594Z Has data issue: false hasContentIssue false
  • English
  • Français

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
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.

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

[1] Bochner, S. and Montgomery, D.: Groups of differentiable and real or complex analytic transformations1, Ann. of Math., vol. 46 (1945), pp. 685–694.Google Scholar
[2] Cartan, E.: Leçons sur la géométrie des espaces de Riemann, Paris, (1946).Google Scholar
[3] Chevalley, C.: Theory of Lie groups I, Princeton University Press, (1946).Google Scholar
[4] Rham, G. de: Sur la réductibilité d’un espace de Riemann, Comm. Math. Helv., vol. 26 (1952), pp. 328–344.CrossRefGoogle Scholar
[5] Hano, J. and Morimoto, vA.: Note on the group of affine transformations of an affinely connected manifold, Nagoya Math. Jour., vol. 8 (1955), pp. 85–95.Google Scholar
[6] Kobayashi, S.: Groupe de transformations qui laissent invariante une connexion infinitésimale, Comptes rendus, 238 (1954), pp. 644–645.Google Scholar
[7] Kobayashi, S.: Espaces à connexion de Cartan complets, Proc. of the Jap. Acad., vol. 30 (1953), pp. 709–710.Google Scholar
[8] Myers, S. and Steenrod, N.: The group of isometries of a Riemannian manifold, Ann. of Math., vol. 40 (1939), pp. 400–416.Google Scholar
[9] Nomizu, K.: On the group of affine transformations of an affinely connected manifold, Proc. of the Amer. Math. Soc., vol. 4 (1953), pp. 816–823.CrossRefGoogle Scholar
[10] Nomizu, K.: Invariant affine connections on homogeneous spaces, Amer. Jour, of Math., vol. 76 (1954), pp. 33–65.CrossRefGoogle Scholar
[11] Nomizu, K.: Sur les transformations affines d’une variété riemannienne, Comptes rendus, 237 (1953), pp. 1308–1310.Google Scholar
[12] Nomizu, K.: Studies on Riemannian homogeneous spaces, In this Journal.Google Scholar
[13] Yano, K.: On harmonic and Killing vector fields, Ann. of Math., vol. 55 (1952), pp. 38–45.Google Scholar