Exploring the Fubini-Study Metric on the 2-Sphere

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In summary, 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. This metric is called the Fubini-Study metric and is defined on CP^n, with CP^1 being equivalent to S^2. The standard metric of constant Gauss curvature can be computed from the unit sphere in Euclidean 3 space by the pullback to R^2 via stereographic projection. This is a good example of Gauss curvature as a conformal invariant.
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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.
 
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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?
 
  • #4
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.
 
  • #5
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.
 
  • #6
This metric is called the Fubini-Study metric. It is defined in general on CP^n, but CP^1=S^2.
 
  • #7
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.
 

What is the 2-sphere?

The 2-sphere, also known as the unit sphere, is a geometric object in three-dimensional space that is defined as the set of all points equidistant from a given point, called the center. It is a two-dimensional surface that can be thought of as the surface of a ball.

What are metrics on the 2-sphere?

Metrics on the 2-sphere refer to mathematical functions that define the distance between two points on the surface of the 2-sphere. These metrics are used to measure the curvature and other geometric properties of the 2-sphere.

Why are metrics on the 2-sphere important?

Metrics on the 2-sphere are important in various fields of science, including geometry, physics, and cosmology. They allow us to quantify and understand the curvature and other geometric properties of the 2-sphere, which is a fundamental concept in many areas of mathematics and science.

How are metrics on the 2-sphere calculated?

The most commonly used metric on the 2-sphere is the Riemannian metric, which is calculated using the Pythagorean theorem. Other metrics, such as the great circle distance, can also be used to measure distances on the 2-sphere.

What are some applications of metrics on the 2-sphere?

Metrics on the 2-sphere have numerous applications, including in navigation systems, computer graphics, and cosmology. They are also used in the study of curved spaces and in the development of Einstein's theory of general relativity.

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