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Transformation Groups with (n-1)-Dimensional Orbits on Non-Compact Manifolds

Published online by Cambridge University Press:  22 January 2016

Tadashi Nagano
Tadashi Nagano*
Affiliation:
Mathematical Institute, University of Tokyo
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When a Lie group G operates on a differentiable manifold M as a Lie transformation group, the orbit of a point p in M under G, or the G-orbit of p, is by definition the submanifold G(p) = {G(p); g∈G}. The purpose of this paper is to characterize the structure of a non-compact manifold M such that there exists a compact orbit of dimension (n — 1), n — dim M, under a connected Lie transformation group G, which is assumed to be compact or an isometry group of a Riemannian metric on M.

Information

Type
Research Article
Information
Nagoya Mathematical Journal , Volume 14 , February 1959, pp. 25 - 38
Copyright
Copyright © Editorial Board of Nagoya Mathematical Journal 1959

References

[ 1 ] Freudenthal, H., Neuere Fassungen des Riemann-Helmholtz-Lieschen Raumproblems. Math. Zeit, 63 (1955-1956), 374–405.CrossRefGoogle Scholar
[ 2 ] Koszul, J.L., Sur certains groupes de transformations de Lie, Colloque de géométrie différentielle, Strasbourg, 1953, 137–141.Google Scholar
[ 3 ] Montgomery, D., Samelson, H. and Yang, C. T., Groups on En with (n — 2)-dimensional orbits. Proc. Amer. Math. Soc, 85 (1956), 719–728.Google Scholar
[ 4 ] Montgomery, D. and Zippin, L.. Topological transformation groups, Interscience Press, 1955.Google Scholar
[ 5 ] Mostert, P.S., On a compact Lie group acting on a manifold. Ann. of Math., 65 (1957), 447-455; 85 (1957), p. 589.CrossRefGoogle Scholar
[ 6 ] Nagano, T., Homogeneous sphere bundles and the isotropic Riemannian spaces, to appear.Google Scholar
[ 7 ] Nagano, T., Compact transformation groups on the n-sphere with (n — l)-dimensional orbits, to appear.Google Scholar
[ 8 ] Tits, J. Sur certaines classes d’espaces homogènes de groupes de Lie, Académie Royale de Belgique. Mémoire t. 85 (1955).Google Scholar
[ 9 ] Wang, H.C., Two-point homogeneous spaces. Ann. of Math., 85 (1952), 177–191.CrossRefGoogle Scholar
[10] Yano, K., The theory of Lie derivatives and its applications. North-Holland Publishing Co. 1957.Google Scholar