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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
DOI: https://doi.org/10.1090/S0002-9947-1975-0423388-4
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