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Reduction Theorem for Connections and its Application to the Problem of Isotropy and Holonomy Groups of a Riemannian Manifold

Published online by Cambridge University Press:  22 January 2016

Katsumi Nomizu
Katsumi Nomizu*
Affiliation:
Mathematical Institute, Nagoya University
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The present paper constitutes, together with [13], a continuation of the study of differential geometry of homogeneous spaces which we started in [11]. Our main result states that if the homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at each point, then the Riemannian manifold is symmetric. The converse is of course one of the well known properties of a Riemannian symmetric space [4]. Besides the results already sketched in [12], we add a few applications of the main theorem.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 9 , October 1955 , pp. 57 - 66
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1955

References

[1] Ambrose, W. and Singer, I., A theorem on holonomy, Trans. Amer. Math. Soc., 75 (1953), 428–443.Google Scholar
[2] Bochner, S., Vector fields and Ricci curvature, Bull. Amer. Math. Soc., 52 (1946), 776–797.CrossRefGoogle Scholar
[3] Cartan, E., Les groupes d’holonomie des espaces généralisés, Acta Math., 48 (1926), 1–42.Google Scholar
[4] Cartan, E., Sur une classe remarquable d’espaces de Riemann, Bull. Soc. Math. France, 54 (1926), 214–264, and 55 (1927), 114–134.Google Scholar
[5] Chern, S. S., Topics in differential geometry, Institute for Advanced Study, 1951.Google Scholar
[6] Chevalley, C., Theory of Lie groups I, Princeton 1946.Google Scholar
[7] Ehresmann, C., Les connexions infinitésimales dans un espace fibré différentiable, Colloque de Topologie, Bruxelles (1950), 29–55.Google Scholar
[8] Kobayashi, S., Des groupes linéaires irréductibles et la géométrie différentielle, Proc. Japan Acad., 30 (1954), 934–936.Google Scholar
[9] Lichnerowicz, A., Espaces homogènes kâhlériens, Colloque de Géométrie Différentielle, Strasbourg (1953), 171–184.Google Scholar
[10] Nomizu, K., On the group of affine transformations of an affinely connected manifold, Proc. Amer. Math. Soc., 4 (1953), 816–823.CrossRefGoogle Scholar
[11] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. Journ. Math., 76 (1954), 33–65.CrossRefGoogle Scholar
[12] Nomizu, K., Remarques sur les groupes d’holonomie et d’isotropie, Colloque de Topologie, Strasbourg (mai, 1954).Google Scholar
[13] Nomizu, K., Studies on Riemannian homogeneous spaces, in this journal.CrossRefGoogle Scholar
[14] Rham, G. de, Sur la réductibilité d’un espace de Riemann, Comment. Math. Helv., 26 (1952), 328–244.CrossRefGoogle Scholar
[15] Steenrod, N., The topology of fiber bundles, Princeton, 1951.CrossRefGoogle Scholar