Lagmental Vicfred

The Upper Half-Plane Metric Is Möbius Invariant by Vicfred

tags: mathematics, Lie groups, hyperbolic geometry, Mobius transformations

The hyperbolic metric divides Euclidean length by the height above the boundary. This is a compact note, but the quantifiers and hypotheses stay on the page.

Statement

A Lie group \(G\) is simultaneously a smooth manifold and a group. Hyperbolic metrics on \(\mathbf H\) and \(\mathbf D\) have Lie groups of Möbius isometries.

$$ \mathbf H=\{z=x+iy:y>0\},\qquad ds^2=\frac{dx^2+dy^2}{y^2} $$

The formulas should not be merged too early. The datum \(\mathsf D\), the conclusion \(\mathsf C\), and the bridge \(\Longrightarrow\) have three different logical jobs.

$$ \gamma z=\frac{az+b}{cz+d},\quad\gamma\in\operatorname{PSL}_2(\mathbf R)\Longrightarrow ds^2(\gamma z)=ds^2(z) $$

Worked algebra

An explicit case prevents the notation from becoming ceremonial. Every subscript and superscript in the display contributes to the value.

$$ \operatorname{Im}(\gamma z)=\frac{\operatorname{Im}z}{|cz+d|^2},\qquad d(\gamma z)=\frac{|dz|}{|cz+d|^2} $$

Conceptual compression

The two-row display is also a debugging tool: if the conclusion changes when only notation changes, some hidden choice has entered the argument.

$$ \begin{aligned} \mathsf{D}\;&:\quad \mathbf H=\{z=x+iy:y>0\},\qquad ds^2=\frac{dx^2+dy^2}{y^2},\\[5pt] \mathsf{C}\;&:\quad \gamma z=\frac{az+b}{cz+d},\quad\gamma\in\operatorname{PSL}_2(\mathbf R)\Longrightarrow ds^2(\gamma z)=ds^2(z). \end{aligned} $$

Caveat

Euclidean drawings distort hyperbolic distance and angle behavior unless the chosen model is conformal. Boundary points are not interior points at finite distance.

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] \gamma z=\frac{az+b}{cz+d},\quad\gamma\in\operatorname{PSL}_2(\mathbf R)\Longrightarrow ds^2(\gamma z)=ds^2(z) \end{gathered}} $$

The important habit is to remember what was fixed before the calculation began and what was proved only afterward. The final display preserves that order.

This article was posted on Wed 21 January 2009. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.