# Standard reference for equivalence of PU(2) action on $\mathbb{C}\mathbb{P}^1$ and SO(3) action on $S^2$

The equivalence I describe below is well-known, but I'd like a simple standard reference for it.

Consider $\mathbb{C}\mathbb{P}^1$, the set of one-dimensional subspaces of $\mathbb{C}^2$, which has a metric given by the angle between subspaces (varying between a minimum of $0$ for identical subspaces and a maximum of $\frac\pi2$ for a subspace and its unique orthogonal complement) and which has holomorphic isometry group $\mathrm{PU}(2)$. Consider on the other hand $\frac12 S^2$, the sphere of points distance $\frac12$ from the origin in $\mathbb{R}^3$, which has a metric given by great-circle distance (varying between a minimum of $0$ for identical points and a maximum of $\frac\pi2$ for a point and its unique antipode) and which has orientation-preserving isometry group $\mathrm{SO}(3)$.

Now define a map $\varphi : \mathbb{C}\mathbb{P}^1 \to \frac12 S^2$. The subspace spanned by $(0,1)$ is sent by $\varphi$ to the north pole $p = (0,0,\frac12)$. Any other subspace is spanned by a uniquely defined vector $(1,a+bi)$, for $a$ and $b$ real and $i^2 = -1$, and $\varphi$ sends it to the point at which the open ray from $p$ through $(a, b, -\frac12)$ intersects $\frac12 S^2$. (This is a shift of the standard stereographic projection to place the center of the sphere at the origin.)

Claim: The map $\varphi$ is an isometry from $\mathbb{C}\mathbb{P}^1$ to $\frac12 S^2$, and the map from $f \in \mathrm{SO}(3)$ to $g = \varphi^{-1} f \varphi \in \mathrm{PU}(2)$ is an isomorphism of Lie groups.

The fact that the two Lie groups are isomorphic is mentioned (without reference, by a sequence of isomorphisms) in Wikipedia and the isometry also appears as a special case of something more specialized. I expect that some version of the equivalence I want is covered in any standard text on quantum computing, where $\mathbb{C}\mathbb{P}^1$ is called the Bloch sphere. If possible I would prefer not to use such specialized references for what is essentially a simple (but somewhat tedious to verify) piece of geometry.

Is there a good standard reference, ideally requiring minimal background beyond standard undergraduate mathematics, that would suffice to treat a collection of vectors in $\mathbb{C}^2$, considered up to individual scaling and simultaneous action by $\mathrm{U}(2)$, as being equivalent (under an explicit map) to a collection of points in a $2$-sphere, considered up to Euclidean geometry?

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I guess a simpler title would have been "Standard reference for the Hopf fibration". –  Tracy Hall Mar 29 '11 at 1:15
I hope that you find the following two references useful 1. Mason A. Porter citeseerx.ist.psu.edu/viewdoc/summary?doi=10.1.1.54.4030 2.Ira S. Moskowitz dtic.mil/cgi-bin/… –  David Bar Moshe Mar 29 '11 at 10:15