Lagmental Vicfred

The Poincaré Disk and Half-Plane Are Isometric by Vicfred

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

A Cayley transform carries the upper half-plane metric to the disk metric. I will separate the object being defined from the consequence being claimed.

Set-up

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.

$$ C(z)=\frac{z-i}{z+i}:\mathbf H\to\mathbf D $$

I keep the defining relation \(\mathsf D\) above the derived relation \(\mathsf C\). This exposes whether cancellation used \(x\ne0\) and whether the conclusion is canonical.

$$ ds_{\mathbf D}^2=\frac{4|dw|^2}{(1-|w|^2)^2} $$

The calculation

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

$$ C^{-1}(w)=i\frac{1+w}{1-w},\qquad\frac{4|dC(z)|^2}{(1-|C(z)|^2)^2}=\frac{|dz|^2}{(\operatorname{Im}z)^2} $$

What survives abstraction

The compact alignment is a local map of the argument: assumptions on the first row, consequence on the second. Any generalisation must preserve that dependency.

$$ \begin{aligned} \mathsf{D}\;&:\quad C(z)=\frac{z-i}{z+i}:\mathbf H\to\mathbf D,\\[5pt] \mathsf{C}\;&:\quad ds_{\mathbf D}^2=\frac{4|dw|^2}{(1-|w|^2)^2}. \end{aligned} $$

The boundary

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

$$ \boxed{\begin{gathered} \text{compact conclusion}\\[-2pt] ds_{\mathbf D}^2=\frac{4|dw|^2}{(1-|w|^2)^2} \end{gathered}} $$

With the dependency made explicit, the same pattern can be recognised safely in nearby problems. A changed hypothesis should now be easy to spot.

This article was posted on Wed 24 January 2018. Facts and circumstances may have changed since publication.
Please contact me before jumping to conclusions if something seems wrong or unclear.