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A Note on Infinitesimal Holonomy Groups*

Published online by Cambridge University Press:  22 January 2016

Albert Nijenhuis
Albert Nijenhuis*
Affiliation:
University of Washington, Seattle, Washington, U.S.A.
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In a recent paper [1], H. Ozeki has extended the author’s results [2] on local and infinitesimal holonomy groups for connections in linear fiber bundles (whose fiber is a vector space) to general fiber bundles whose structure group is a Lie group. Ozeki’s Lemma 7 (corresponding to the author’s Lemma 5.2) appears to be rather crucial in the development, while the proof is somewhat involved.—This note intends to present a more direct proof of the lemma, stating a case in which the local holonomy group H*(x)and the infinitesimal holonomy group H’(x)coincide:

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 12 , December 1957 , pp. 145 - 146
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1957

Footnotes

*

Research sponsored by the Office of Naval Research.

References

Literature

[1] Ozeki, H.. Infinitesimal holonomy groups of hundle connections. Nagoya Math. J., 10, 105–123 (1956).CrossRefGoogle Scholar
[2] Nijenhuis, A.. On the holonomy groups of linear connections. Kon. Nederl. Akad. Wetenschappen Amsterdam, Proc, A56 (3), 233–249, A57 (1), 17–25 (=Indag. Math., 15 (3), 233–249; 16 (1), 17–25).Google Scholar