The FubiniStudy metric d(x,y) on $CP^1$ is defined as follows: for x and y in $CP^1$ let v and w be unit vectors in $C^2$ representing x and y. Then $d(x,y)=2arccos(\langle v,w\rangle)$. The round metric on $S^2$ is given by taking a pair of points to the angle between those two points. It is a fact that under stereographic projection the round metric corresponds to the FubiniStudy metric. This can be proven by some complicated and unenlightening algebraic manipulation. On the other hand, it seems like maybe there should be a purely (or almost purely) geometric proof of this fact for the following reason. Embed $S^2$ into $C^2$ by taking it to the unit sphere around (0,0,1,0) in v=0, where $C^2$ has coordinates z=x+iy and w=u+iv,so the north pole of $S^2$ is just the origin (0,0,0,0). Then the for two points a and b on the sphere, the angle metric is the angle aCb where C is the center of the sphere, (0,0,1,0). On the other hand, the FubiniStudy metric is just $2arccos(\langle x,y\rangle/\mid x\mid \mid y\mid)$ , which, if this were the ordinary Euclidean inner product, would just be twice the angle aOb where O is the origin (0,0,0,0). The statement that these two are equal is highly reminiscent of the highschool geometry fact that the angle measure between two points on a circle measured from the center is twice the angle measure between two points measured from another point on the circle. Of course, this is not a circle, and the inner product is not the Euclidean inner product. Is the similarity just a coincidence, or is there some geometric fact that explains what's going on?
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