How Does the Fubini-Study Metric Compare to the Standard Metric on the 2-Sphere?

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Discussion Overview

The discussion revolves around the comparison of the Fubini-Study metric and the standard metric on the 2-sphere, focusing on their properties, particularly Gauss curvature. Participants explore the mathematical definitions and implications of these metrics within the context of Riemannian geometry.

Discussion Character

  • Technical explanation
  • Debate/contested
  • Mathematical reasoning

Main Points Raised

  • One participant presents the metric ds = |dz|/(1 + |z|^2) and claims it has constant positive Gauss curvature equal to 4, questioning its relationship to the standard metric derived from the unit sphere in Euclidean 3 space.
  • Another participant suggests a possible misunderstanding, proposing that the metric might actually be ds = |dz|/(1 + |z|^2)^2, which they claim represents one fourth of the standard metric.
  • A participant expresses uncertainty about the metric's correctness and seeks clarification on how the alternative metric is considered the standard one.
  • One participant explains that the standard metric can be obtained by computing the pullback to R^2 of the round metric via stereographic projection, referencing a specific source for this computation.
  • Another participant agrees with the computation and notes that both metrics discussed are indeed the same, clarifying a potential misunderstanding regarding the notation used.
  • A participant identifies the metric as the Fubini-Study metric, noting its general definition on CP^n and its specific case for CP^1 as the 2-sphere.
  • One participant mentions confusion regarding the inverse tangent in the metric's definition but acknowledges understanding its role in determining radial length.

Areas of Agreement / Disagreement

Participants express differing views on the correct form of the metric and its implications for Gauss curvature. There is no consensus on the relationship between the Fubini-Study metric and the standard metric, as some participants propose alternative interpretations and calculations.

Contextual Notes

Limitations include potential misunderstandings regarding the definitions of the metrics and the assumptions underlying the computations. The discussion does not resolve the discrepancies in the metric forms or their implications for curvature.

lavinia
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The metric ds = |dz|/(1 + |z|^2) has constant positive Gauss curvature equal to 4 and extends to the complex plane plus the point at infinity. How does this metric relate to the usual metric of constant Gauss curvature computed from the unit sphere in Euclidean 3 space?
 
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I assume you mean what you wrote, and not ds = |dz|/(1 + |z|^2)^2 in which case that's just one fourth of the standard metric.
 
quasar987 said:
I assume you mean what you wrote, and not ds = |dz|/(1 + |z|^2)^2 in which case that's just one fourth of the standard metric.

I will have to think about this. I am pretty sure that I gave the right metric. How is the one you gave the standard one?
 
By computation of the pullback to R^2 of the round metric by stereographic projection. It is done in the book by John M. Lee (Riemannian manifolds) at p.37.
 
quasar987 said:
By computation of the pullback to R^2 of the round metric by stereographic projection. It is done in the book by John M. Lee (Riemannian manifolds) at p.37.

Thanks quasar. The computation is straight forward. BTW we have the same metric. you were thinking of ds^2.

This is a good example of Gauss curvature as a conformal invariant.
 
This metric is called the Fubini-Study metric. It is defined in general on CP^n, but CP^1=S^2.
 
quasar987 said:
This metric is called the Fubini-Study metric. It is defined in general on CP^n, but CP^1=S^2.

Thanks.

The thing that threw me was the inverse tangent but I now see why that gives the radial length.
 

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