I’m looking over some fairly basic stuff on complex methods and the book I’m using takes the formula for the stereographic projection: $$z = \cot(\beta/2)e^{i\phi} $$ as well as the line element on the sphere: $$ ds^2 = d\beta^2 + \sin^2 \beta d\phi^2 $$ (where $\beta$ is the polar angle and $\phi$ the azimuthal angle) […]

Background: Let $S^2$ denote the unit sphere in $\mathbb{R}^3$. By “stereographic projection”, I mean the mapping from $S^2$ (remove the north pole) to the complex plane which sends \begin{align*} \begin{bmatrix} x \\ y \\ t \end{bmatrix} \in S^2 \mapsto z = \frac{x+iy}{1-t} \in \mathbb{C}. \end{align*} The inverse mapping, from the complex plane to the sphere, […]

This previous question had me thinking about something I’ve taken for granted. Consider $\mathrm{SL}_2(\mathbb{C})$ acting on $\mathbb{C}^2$, which descends to an action on the complex projective line $\mathbb{CP}^1$, which we may think of as the Riemann sphere $\widehat{\mathbb{C}}=\mathbb{C}\cup\{\infty\}$. It acts by Möbius transformations. By stereographic projection, $\widehat{\mathbb{C}}$ may be identified with a literal sphere $\mathbb{S}^2\subset […]

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