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A Characterization of Invariant Affine Connections

Published online by Cambridge University Press:  22 January 2016

Bertram Kostant
Bertram Kostant*
Affiliation:
University of California
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1. Introduction and statement of theorem. 1. In [1] Ambrose and Singer gave a necessary and sufficient condition (Theorem 3 here) for a simply connected complete Riemannian manifold to admit a transitive group of motions. Here we shall give a simple proof of a more general theorem — Theorem 1 (the proof of Theorem 1 became suggestive to us after we noted that the Tx of [1] is just the ax of [6] when X is restricted to p0, see [6], p. 539).

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 16 , February 1960 , pp. 35 - 50
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1960

References

[1] Ambrose, W. and Singer, I. M., On homogeneous Riemannian manifolds, Duke Mathematical Journal, 25 (1950), pp. 647–669.Google Scholar
[2] Hano, J. and Morimoto, A., Note on the group of affine transformations of an affinely connected manifold, Nagoya Mathematical Journal, 8 (1955), pp. 71–81.Google Scholar
[3] Hano, J., On affine transformations of a Riemannian manifold, Nagoya Mathematical Journal, 9 (1955), pp. 99–109.CrossRefGoogle Scholar
[4] Kobayashi, S., Espaces à connexions affine et Riemanniennes symétriques, Nagoya Mathematical Journal, 9 (1955), pp. 25–37.Google Scholar
[5] Kobayashi, S., A theorem on the affine transformation group of a Riemannian manifold, Nagoya Mathematical Journal, 9 (1955), pp. 39–41.CrossRefGoogle Scholar
[6] Kostant, B., Holonomy and the Lie algebra of infinitesimal motions of a Riemannian manifold, Transactions of the American mathematical Society, 80 (1955), pp. 528–542.CrossRefGoogle Scholar
[7] Nomizu, K., Invariant affine connections on homogeneous spaces, American Journal of Mathematics, 76 (1954), pp. 33–65.Google Scholar
[8] Nomizu, K., Lie groups and differential geometry. Publications of the Mathematical Society of Japan, Tokyo, 1956.Google Scholar