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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

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