The action of SU(2) on the Riemann sphere

Question

One way to get the famous double cover $\text{SU}(2) \to \text{SO}(3)$ is to note that $\text{SU}(2)$ is isomorphic to the group of unit quaternions and to let unit quaternions $q$ act on the subspace $V$ of $\mathbb{H}$ spanned by $i, j, k$ via conjugation $t \mapsto qtq^{-1}$; this preserves the norm. (Alternately, this is the adjoint action on the Lie algebra, which preserves the Killing form.)

Another way to do this is to let $\text{SU}(2)$ act on $\mathbb{P}^1(\mathbb{C})$, which is diffeomorphic to the sphere. This gives an action of $\text{SU}(2)$ by conformal automorphisms. However, I don't know how to prove that $\text{SU}(2)$ actually acts by rotations (at least, not without some explicit and unenlightening calculations).

To be more precise, if we fix an inner product on $\mathbb{C}^2$, then the space of lines in $\mathbb{C}^2$ can be given the Fubini-Study metric, which $\text{SU}(2)$ preserves. But how can I prove that the Fubini-Study metric agrees with the natural metric on the sphere (up to a constant)?

Answer 1

With the F-S metric, $\mathbf P^1(\mathbb C)$ is a Riemannian surface upon which a group acts transitively. That implies that the curvature is constant. Now, the classification of space forms, for example, shows that such a thing is covered locally isometrically by $S^2$, the round sphere, $E^2$, the flat plane, or $H^2$, the hyperbolic plane. Since $\mathbf P^1(\mathbb C)$ is simply connected and compact, its only covering is the identity, and since it is compact, it must be a round sphere.

Answer 2

Okay, forget the Fubini-Study metric. I think I have an alternate solution. Let's rephrase the problem as follows. We'll pick a local coordinate $z$ and think of elements of $\text{PSL}_2(\mathbb{C})$ as fractional linear transformations $z \mapsto \frac{az + b}{cz + d}$. Adding an inner product on the underlying copy of $\mathbb{C}^2$ lets us associate to any $z$ its "orthogonal complement" $- \frac{1}{\bar{z}}$, and the right choice of stereographic projection sends orthogonal complements to antipodes.

Now, an element of $\text{PSL}_2(\mathbb{C})$ respects antipodes (equivalently, orthogonal complements) if and only if it lies in $\text{PSU}(2)$, so it remains to show that any conformal orientation-preserving antipode-preserving automorphism of the Riemann sphere is a rotation. Certainly a fractional linear transformation $g$ preserving antipodes must have two antipodal fixed points of the same type. They can't both be attractive or both repelling (that contradicts the fact that the product of the eigenvalues is $1$), so they are both neither. This ought to already be enough to conclude that $g$ preserves distances to its fixed points, which should only be possible if it's a rotation (and expanding at a local coordinate at the fixed points shows this if nothing else does).