# 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

## 1 Answer

I didnt enter into the details of your question, but it seems that a natural answer would be: CP1 is a homogeneous space SU(2)/U(1) meaning that there is a transitive action of SU(2) with isotropy U(1). On the other hand SO(3) acts on S2 by rotations with isotropy SO(2). So you need to pass from PSU(2) to SO(3). This is a classical trick which comes from Gauss: any quaternion q of norm 1 parametrizes a rotation in R3 because the map x \mapsto qxq^{-1} from H=R4 to H=R4 preserves the norm, hence the scalar product. This is done in many books under the name: SU(2) is the universal covering of SO(3). The difficult thing is to prove that any rotation can be parametrized in this way: one possibility is to express q in the form sinz+cosz w, then you obtain the rotation with angle 2z and rotation axis w...

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