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Imagine two smooth scalar functions defined on the surface of a sphere. Is it possible to find a smooth coordinate translation that slides the values of the first function over the sphere, until it becomes equal to the second function everywhere?

It seems reasonable that if the number of maxima and minima are different, then we can't stretch one function into another. But if the number of maxima and minima are the same, I'm not sure if it's always possible.

[Edit: Oops ... we need to assume that each maximum/minimum of f1 has the same value as a corresponding maximum/minimum of f2, although in a possibly different location.] Let's assume that as well.

My questions are, which branch(es) of mathematics address(es) this kind of thing? What are some key results that exist for this topic? How does it apply to the previous paragraphs?

It seems reasonable that if the number of maxima and minima are different, then we can't stretch one function into another. But if the number of maxima and minima are the same, I'm not sure if it's always possible.

[Edit: Oops ... we need to assume that each maximum/minimum of f1 has the same value as a corresponding maximum/minimum of f2, although in a possibly different location.] Let's assume that as well.

My questions are, which branch(es) of mathematics address(es) this kind of thing? What are some key results that exist for this topic? How does it apply to the previous paragraphs?

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