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

[lavinia]

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?

 

[quasar987]

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.

 

[lavinia]

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?

 

[quasar987]

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.

 

[lavinia]

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.

 

[quasar987]

This metric is called the Fubini-Study metric. It is defined in general on CP^n, but CP^1=S^2.

 

[lavinia]

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.