What Is the Usual Metric on the Sphere That do Carmo Doesn't Mention?

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The discussion centers on the concept of the "usual metric" on the sphere in relation to the antipodal mapping, specifically addressing why the author, do Carmo, does not explicitly mention this metric in his problem statement. It is established that the antipodal mapping A: S^n --> S^n defined by A(p) = -p is indeed an isometry under the usual metric on the sphere. Furthermore, it is clarified that the antipodal mapping does not qualify as an isometry for any other metric on the sphere, emphasizing the importance of assuming the usual metric unless stated otherwise.

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InbredDummy
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"Usual metric"

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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Well, I can't speak for Carmo but, yes, for problems in Rn or subsets or Rn, as here, unless a different metric is specified, assume the usual metric.
 
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is your friend given to habitual "one -ups manship"?

next time he says something about zero loci, ask if he means them to have their induced reduced scheme structure?
 

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