complex-analysis
Out of curiosity, is there a nice characterization of the linear fractional transformations which give rotations of the Riemann sphere?
My thinking was a rotation of the Riemann sphere rotates about some axis, and the two points where the sphere intersects the axis will be two fixed points. What more be said of this?
The Riemann sphere is not just $\mathbb{C} \cup \{ \infty\}$. It this space endowed with a particular topology. You can think of that topology as arising from adding infinities at the end of some "infinitely large" circle, and then collapsing all those infinities to a point. This perspective is highlighted when considering the stereographic projection of $S^2$ onto $\mathbb{R}^2 $ or $ \mathbb{C}$.
Horizontal circles wrapping around the sphere are mapped to circles in the plane. As we push the circles higher on the Riemann sphere, the corresponding circles in the plane grow. At the north pole, we should have an "infinitely large" circle in the plane, but since the north pole is just a point, that "infinitely large" circle is actually just a point, which we call the point at infinity.
From a technical, point-set topology perspective, this is an instance of the Alexandroff one-point compactification. Another way to view the Riemann sphere is as the complex projective line.
A generic Möbius transformation is a meromorphic function on $\mathbb{C}$, and a holomorphic map of the Riemann sphere onto itself. There is a difference between "function" and "map" here.
When the Riemann sphere is equipped with the structure of a complex manifold, a neighborhood of $\infty$ is covered by the coordinate patch $z\mapsto 1/z$. A function with a pole qualifies as a holomorphic map into the sphere, because at a point $a$ where $f(a)=\infty$, the composition with the above chart map is $1/f$ (which is holomorphic).
Best Answer
Yes, there is! See this for details.
"A map of the Riemann sphere to itself is a rotation if and only if the corresponding map induced on the plane by stereographic projection is a linear fractional transformation whose coefficient matrix is unitary."