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Problem: Suppose that $f:S^n\to S^n$ is a mapping from the n-dimensional sphere ($n\geq 3$) into itself which maps circles into (instead of onto) circles. Can we say that f maps (n-1)-dimensional spheres into (n-1)-dimensional spheres?

Here, we make no any other assumption on f, e.g. continuity, injectivity, surjectivity, and so on. Circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle.

Note that neither of two "into"s in assumptions means "onto".

If you can provide related literature to the problem, I will be very grateful.

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Problem: Suppose that $f:S^n\to S^n$ is a mapping from the n-dimensional sphere ($n\geq 3$) into itself which maps circles into (instead of onto) circles. Can we say that f maps (n-1)-dimensional spheres into (n-1)-dimensional spheres?

Here, we make no any other assumption on f, e.g. continuity, injectivity, surjectivity, and so on. Circle is in the ordinary sense, i.e. round circle (or say 1-sphere), not necessarily great circle.

Note that neither of two "into"s in assumptions means "onto".

If you can provide related literature to the problem or point me to what is the extent of the importance to solve the problem, I will be very grateful.

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