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Transactions of the American Mathematical Society

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

 

 

Circle-preserving functions of spheres


Authors: Joel Gibbons and Cary Webb
Journal: Trans. Amer. Math. Soc. 248 (1979), 67-83
MSC: Primary 51M10
MathSciNet review: 521693
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Abstract: Suppose a function of the standard sphere $ {S^2}$ into the standard sphere $ {S^{2 + m}}$, $ m \geqslant 0$, sends every circle into a circle but is not a circlepreserving bijection of $ {S^2}$. Then the image of the function must lie in a five-point set or, if it contains more than five points, it must lie in a circle together with at most one other point. We prove the local version of this theorem together with a generalization to n dimensions. In the generalization, the significance of 5 is replaced by $ 2n + 1$. There is also proved a 3-dimensional result in which, compared to the n-dimensional theorem, we are allowed to weaken the structure assumed on the image set of the function.


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DOI: https://doi.org/10.1090/S0002-9947-1979-0521693-8
Keywords: Circle, sphere, inversive group
Article copyright: © Copyright 1979 American Mathematical Society