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On Infinitesimal Holonomy and Isotropy Groups

Published online by Cambridge University Press:  22 January 2016

Katsumi Nomizu
Katsumi Nomizu*
Affiliation:
Mathematical Institute, Nagoya University
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We have proved in [2] that if the restricted homogeneous holonomy group of a complete Riemannian manifold is contained in the linear isotropy group at every point, then the Riemannian manifold is locally symmetric, that is, the covariant derivatives of the curvature tensor field are zero. The proof of this theorem, however, depended on an insufficiently stated proposition (Theorem 1, [2]). In the present note, we shall give a proof of a more general theorem of the same type.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 11 , February 1957 , pp. 111 - 114
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

References

[1] Nijenhuis, A., On the holonomy groups of linear connections, I, II, Indagationes Math. 56 (1953), pp.233–249.Google Scholar
[2] Nomizu, K., Reduction theorem for connections and its application to the problem of isotropy and holonomy groups of a Riemannian manifold, Nagoya Math. Journ. 9 (1955), pp.57–66.Google Scholar
[3] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. Journ. Math. 76 (1954), pp.33–65.Google Scholar
[4] Nomizu, K., Lie groups and differential geometry Publications of the Mathematical Society of Japan, No. 2 (1956).Google Scholar