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Note on the Group of Affine Transformations of an Affinely Connected Manifold

Published online by Cambridge University Press:  22 January 2016

Jun-Ichi Hano  and
Akihiko Morimoto
Jun-Ichi Hano
Affiliation:
Mathematical Institute, Nagoya University
Akihiko Morimoto
Affiliation:
Mathematical Institute, Nagoya University
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The purpose of the present note is to reform Mr. K. Nomizu’s result on the group of all affine transformations of an affinely connected manifold. We shall prove the following.

THEOREM. The group of all affine transformations of an affinely connected manifold is a Lie group.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 8 , February 1955 , pp. 71 - 81
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

1) Nomizu, K.; On the group of affine transformations of an affinely connected manifold, Proc. Amer. Math. Soc. vol. 4 (1953).CrossRefGoogle Scholar

2) For the definition of “completeness” see.1)

3) Kobayasi, S.; Groupe de transformations qui laissent invariante une connexion infinitesimale, Comptes rendus, 238 (1954).Google Scholar

4) The term “differentiable” will always mean “of class C∞ “. As for the definitions and the notations of manifold, tangent vector, differential form, etc. we follow Chevalley, C.; Theory of Lie groups, Princeton University Press, 1946 Google Scholar. A manifold is not necessarily connected.

5) cf. 1) or Chern, S.; Lecture note on differential geometry, Princeton.Google Scholar

6) Hereafter we consider geodesic curve always with the canonical parameter.

7) When we say simply that N is a regular neighbourhood of p it means N is a regular neighbourhood of p contained in some neighbourhood U of p. cf. 1).

8) cf.Myers, S. and Steenrod, N.; The group of isometres of a Riemannian manifold, Ann. of Math. vol. 40 (1939).CrossRefGoogle Scholar

9) cf. S. Chern’s, lecture note in 5K

10) cf. S. Chern’s lecture note in 5).

11) Dantzig, van und Waerden, van der, Über metrischen homogene Räume, Abh. Math. Sem. Hamburg, vol. 6 (1928).CrossRefGoogle Scholar

12) cf. 1).

13) Bochner, S. and Montgomery, D.; Locally compact groups of differentiable transformations, Ann. of Math. vol. 47 (1946).CrossRefGoogle Scholar

14) Kuranishi, M.; On conditions of differentiability of locally compact groups, Nagoya Math. J. vol. 1 (1950).CrossRefGoogle Scholar