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Infinitesimal Holonomy Groups of Bundle Connections

  • Hideki Ozeki (a1)

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In Introduction In differential geometry of linear connections, A. Nijenhuis has introduced the concepts of local holonomy group and infinitesimal holonomy group and obtained many interesting results [6].

The purpose of the present note is to generalize his results to the case of connections in arbitrary principal fiber bundles with Lie structure groups. The concept of local holonomy group can be immediately generalized and has been already utilized by S. Kobayashi [4]. Our main results are Theorems 4 and 5 on infinitesimal holonomy groups. The proofs depend on a little sharpened form of a theorem of Ambrose-Singer [1]. In the case of linear connections, our infinitesimal holonomy group coincides with that of Nijenhuis, as we shall show in Section 6.

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References

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[ 1 ] Ambrose, W. and Singer, I., A theorem on holonomy, Trans. Amer. Math. Soc., 75 (1953), 428443.
[ 2 ] Chevalley, C., Theory of Lie groups I, Princeton 1946.
[ 3 ] Ehresmann, C., Les connexions infinitésimales dans un espace fibré differentiable, Colloque de Topologie, Bruxelles (1950), 2955.
[ 4 ] Kobayashi, S., Holonomy groups of hyper surfaces, in this Journal.
[ 5 ] Kobayashi, S., Theory of connections, part I, Mimeographed notes, University of Washington, Seattle, 1955.
[ 6 ] Nijenhuis, A., On the holonomy groups of linear connections, Koninkl. Nederl. Akademie van Wetenschappen, Amsterdam, Proc, Series A, 56 (1953), 233249. A, 57 (1954), 1725.
[ 7 ] Nijenhuis, A., Abstract, No. 334 t, Bull. Amer. Math. Soc. vol. 61, (1955), p. 169.
[ 8 ] Nomizu, K., Lie groups and Differential Geometry, No. 2, Publications Math. Soc. Japan, to appear shortly.
[ 9 ] Nomizu, K., Invariant affine connections on homogeneous spaces, Amer. Jour. Math., 76 (1954), 3365.
[10] Nomizu, K., Reduction theorem for connections and its applications to the problem of isotropy and holonomy groups of Riemannion manifold, Nagoya Math. Jour. 9 (1955), 5766.
[11] Yamabe, H., On arcwise connected subgroups of a Lie group, Osaka Math. Jour. 2 (1950), 1314.
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Nagoya Mathematical Journal
  • ISSN: 0027-7630
  • EISSN: 2152-6842
  • URL: /core/journals/nagoya-mathematical-journal
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