Suppose a function of the standard sphere S2 into the standard sphere S2+m, m > 0, sends every circle into a circle but is not a circlepreserving bijection of S2. Then the image of the function must lie in a five-point set or, if it contains more than five points, it must he 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 In + 1. There is also

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... 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.