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Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)



Newman's theorem in the Riemannian category

Authors: L. N. Mann and J. L. Sicks
Journal: Trans. Amer. Math. Soc. 210 (1975), 259-266
MSC: Primary 57E10
MathSciNet review: 0423388
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Abstract: In 1931 M. H. A. Newman proved that if $ M$ is a connected topological manifold with metric $ d$, there exists a number $ \varepsilon > 0$, depending only upon $ M$ and $ d$, such that every compact Lie group $ G$ acting effectively on $ M$ has at least one orbit of diameter at least $ \varepsilon $. Aside from isolated results nothing appears to be known about $ \varepsilon $. In order to learn more about the invariant $ \varepsilon $, attention is restricted here to groups of isometries on a Riemannian manifold. It is found that the invariant $ \varepsilon $ of $ M$ is connected with the notion of convexity introduced by J. H. C. Whitehead in 1932.

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Keywords: Newman's theorem on periodic transformations, groups of isometries, diameter of orbits, radius of convexity
Article copyright: © Copyright 1975 American Mathematical Society

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