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A Characterization of Invariant Affine Connections

  • Bertram Kostant
Chapter

Abstract

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 T x of [1] is just the a x of [6] when X is restricted to p 0, see [6], p. 539). In fact after introducing, below, the notion of one affine connection A on a manifold being rigid with respect to another affine connection B on M and making some observations concerning such a relationship, Theorem 1 is seen to be a reformulation of Theorem 2.

Keywords

Riemannian Manifold Mathematical Method Differential Geometry Topological Group Simple Proof 
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References

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

Copyright information

© Springer-Verlag New York 2009

Authors and Affiliations

  1. 1.Department of MathematicsMassachusetts Institute of TechnologyCambridgeUSA

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