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So I had to solve a problem proving that the antipodal mapping on the sphere is an isometry. However, someone told me that the antipodal mapping is an isometry on the "usual metric" on the sphere, and in particular, the antipodal mapping is not an isometry for any metric on the sphere. While this makes intuitive sense, why does do Carmo not mention it in the question? The question says:

"Prove that the antipodal mapping A: S^n --> S^n given by A(p) = -p is an isometry of S^n"

He doesn't say anything about using the "usual metric" sphere. Is it just obvious? Should I always suppose that do Carmo is referring to the "usual metric"?

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# Usual metric

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