Circle-preserving functions of spheres
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- by Joel Gibbons and Cary Webb PDF
- Trans. Amer. Math. Soc. 248 (1979), 67-83 Request permission
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.References
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Additional Information
- © Copyright 1979 American Mathematical Society
- Journal: Trans. Amer. Math. Soc. 248 (1979), 67-83
- MSC: Primary 51M10
- DOI: https://doi.org/10.1090/S0002-9947-1979-0521693-8
- MathSciNet review: 521693