Results 1 to 1 of 1

Math Help - Where does the Poincaré metric come from?

  1. #1
    Newbie
    Joined
    Aug 2010
    Posts
    6

    Where does the Poincaré metric come from?

    According to Poincaré disk model - Wikipedia, the free encyclopedia the transformation map \vec{w} = \left(\begin{array}{c} x_1 \\ x_2 \end{array} \right) from points y_i on the Poincaré disk to points x_i on the hyperboloid is 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  g_{ij} = \langle{\frac{\partial \vec{w}}{\partial y_i}}, {\frac{\partial \vec{w}}{\partial y_j}}\rangle yields

    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}

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

     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:

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

     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...
    Last edited by huldrich; March 23rd 2012 at 12:27 PM.
    Follow Math Help Forum on Facebook and Google+

Similar Math Help Forum Discussions

  1. Poincare Maps
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: March 4th 2011, 11:11 AM
  2. Poincare Map
    Posted in the Differential Equations Forum
    Replies: 3
    Last Post: January 5th 2011, 02:28 PM
  3. Poincare Map
    Posted in the Differential Equations Forum
    Replies: 0
    Last Post: November 2nd 2009, 10:28 PM
  4. Poincaré Conjecture
    Posted in the Math Forum
    Replies: 4
    Last Post: August 24th 2007, 04:46 AM
  5. Poincare conjecture
    Posted in the Math Forum
    Replies: 4
    Last Post: August 21st 2006, 03:31 PM

Search Tags


/mathhelpforum @mathhelpforum