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Holonomy Groups Of Hypersurfaces

Published online by Cambridge University Press:  22 January 2016

Shoshichi Kobayashi
Shoshichi Kobayashi*
Affiliation:
University of Washington
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The restricted homogeneous holonomy group of an n–dimensional Riemannian manifold is a connected closed subgroup of the proper orthogonal group SO(n) [1]. In this note we shall prove that the restricted homogeneous holonomy group of an n-dimensional compact hypersurface in the Euclidean space is actually the proper orthogonal group SO(n) itself. This gives a necessary (of course, not sufficient) condition for the imbedding of an n-dimensional compact Riemannian manifold into the (n +1)–dimensional Euclidean space.

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 10 , June 1956 , pp. 8 - 14
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1956

References

[ 1 ] Ambrose, , W-Singer, I. M., A theorem on holonomy, Trans. Amer. Math. Soc. 75 (1953), p. 428–443.Google Scholar
[ 2 ] Borel, A.-Lichnerowicz, A., Groupes d’holonomie des variétés riemanniennes, C. R. Acad. Sci. Paris 234 (1952), p. 1835–1837.Google Scholar
[ 3 ] Ehresmann, C., Les connexions infinitésimales dans un espace fibré différentiable, Colloque de topologie, Bruxelles (1950).Google Scholar
[ 4 ] Nijenhuis, A., On the holonomy groups of linear connections, Nederl. Akad. Wetensch. Proc. A 56 (1953), p. 233–249.Google Scholar
[ 5 ] Kobayashi, S., Induced connections and imbedded Riemannian spaces, this journal.Google Scholar