# Where does the Poincaré metric come from?

• Mar 23rd 2012, 12:15 PM
huldrich
Where does the Poincaré metric come from?
According to Poincaré disk model - Wikipedia, the free encyclopedia the transformation map $\displaystyle \vec{w} = \left(\begin{array}{c} x_1 \\ x_2 \end{array} \right)$ from points $\displaystyle y_i$ on the Poincaré disk to points $\displaystyle x_i$ on the hyperboloid is $\displaystyle x_i = \frac{2 y_i}{1-y_1^2-y_2^2}$. I verified the calculation, this transformation seems to be correct (without considering the hight t). However, the metric $\displaystyle g_{ij} = \langle{\frac{\partial \vec{w}}{\partial y_i}}, {\frac{\partial \vec{w}}{\partial y_j}}\rangle$ yields

$\displaystyle g_{11} = \frac{4 \left(y_1^4+\left(y_2^2-1\right){}^2+2 y_1^2 \left(y_2^2+1\right)\right)}{\left(1-y_1^2-y_2^2\right){}^4}$

$\displaystyle g_{12} = g_{21} = \frac{16 y_1 y_2}{\left(1-y_1^2-y_2^2\right){}^4}$

$\displaystyle g_{22} = \frac{4 \left(y_1^4+\left(y_2^2+1\right){}^2+2 y_1^2 \left(y_2^2-1\right)\right)}{\left(1-y_1^2-y_2^2\right){}^4}$

which is not the metric mentionned in the article, that is:

$\displaystyle g_{11} = g_{22} = \frac{4}{\left(1-y_1^2-y_2^2\right){}^2}$

$\displaystyle g_{12} = g_{21} = 0$

This is known as the Poincaré disk metric and I would really like to know how to calculate it geometrically from the hyberboloid model as explained in the Wikipedia article. So can anyone explain me what's wrong here? Do I need to still apply another transformation to get the right metric? I tried but this does not seem to be an easy task...