Exploring the "Usual Metric" on the Sphere: What Do Carmo Doesn't Mention

In summary, the conversation discussed the use of the "usual metric" on the sphere in solving a problem involving the antipodal mapping. While it may not have been explicitly mentioned in the question, it is generally assumed to be the metric being referred to unless otherwise specified. The speaker also expressed uncertainty about whether the use of the "usual metric" was obvious or not, and speculated about the intentions of the person asking the question.
  • #1
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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  • #2


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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  • #3


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?
 

1. What is the "usual metric" on the sphere?

The "usual metric" on the sphere refers to the standard way of measuring distances between points on a sphere. It is also known as the great-circle distance or the geodesic distance. It is the shortest distance between two points on the surface of a sphere, and it follows the curve of the sphere's surface.

2. How is the "usual metric" calculated on the sphere?

The "usual metric" on the sphere is calculated using the formula d = r * θ, where d is the distance, r is the radius of the sphere, and θ is the angle between the two points measured in radians. This formula takes into account the curvature of the sphere, unlike the Pythagorean theorem used for measuring distances on a flat surface.

3. What are the applications of the "usual metric" on the sphere?

The "usual metric" on the sphere has various applications in fields such as mathematics, geography, and navigation. It is used to calculate distances between cities or landmarks on the Earth's surface, determine the shortest route for air or sea travel, and measure the curvature of the Earth's surface.

4. How does the "usual metric" on the sphere differ from the "usual metric" on a flat surface?

The "usual metric" on the sphere takes into account the curvature of the surface, while the "usual metric" on a flat surface assumes that the surface is flat. This means that the distance between two points on a sphere will always be greater than the distance between the same two points on a flat surface, even if they are located at the same coordinates.

5. Are there any limitations to using the "usual metric" on the sphere?

Yes, there are limitations to using the "usual metric" on the sphere. It is only accurate for short distances compared to the size of the sphere. For longer distances, the curvature of the sphere becomes more significant, and other methods, such as the Haversine formula, may be used. Additionally, the "usual metric" does not take into account changes in elevation, which can affect the actual distance between two points on the sphere.

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