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Circle-Preserving Functions of Spheres

Joel Gibbons and Cary Webb
Transactions of the American Mathematical Society
Vol. 248, No. 1 (Feb., 1979), pp. 67-83
DOI: 10.2307/1998737
Stable URL: http://www.jstor.org/stable/1998737
Page Count: 17
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Circle-Preserving Functions of Spheres
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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 circle- preserving 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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