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Computing Curvature in 2D

  1. May 1, 2013 #1
    1. The problem statement, all variables and given/known data
    I have a given Metric:
    [itex]ds^{2}=A(u,v)^{2}du^{2}+B(u,v)^{2}dv^{2}[/itex]
    And I'm asked to compute its curvature, and use this result to compute the curvature of the poincare metric:
    Set [itex]A=B=\frac{1}{v^{2}}[/itex]

    3. The attempt at a solution
    I'm using Cartan's method. So first I change to an orthonormal frame:
    [itex]σ^{1}=Adu[/itex] and [itex]σ^{2}=Bdv[/itex]
    First I need to find the unique [itex]w_{12}=-w_{21}[/itex]
    So I let [itex]w_{12}=a(u,v)σ^{1}+b(u,v)σ^{2}[/itex] Where a and b are unknown functions I'm looking for. Next I have:
    [itex]dσ^{1}=-w_{12}[/itex]^[itex]σ^{2}=d(Adu)[/itex]
    and
    [itex]dσ^{2}=w_{12}[/itex]^[itex]σ^{1}=d(Bdv)[/itex]
    from Cartan's 1st Structural equations in orthonormal basis.
    This gives me
    [itex]a=\frac{1}{AB}\frac{dA}{dv}[/itex] and [itex]b=-\frac{1}{AB}\frac{dB}{du}[/itex]

    Then I use Cartan's second structural equation to find the curvature K (which is just the coefficient of [itex]dw[/itex]). I find
    [itex]K=\frac{1}{AB}[\frac{1}{A}\frac{d^{2}B}{du^{2}}-\frac{1}{B}\frac{d^{2}A}{dv^{2}} -\frac{1}{A^{2}}\frac{dA}{du}\frac{dB}{du}+\frac{1}{B^{2}}\frac{dA}{dv}[/itex][itex]\frac{dB}{dv}][/itex]

    But then when I plug in [itex]A=B=\frac{1}{v^{2}}[/itex] I get [itex]-2v^{2}[/itex]

    But the Curvature for the Poincare metrix should be -1 (right?)

    What am I doing wrong? Is it just a computational error?

    Any help would be appreciated!
     
    Last edited: May 1, 2013
  2. jcsd
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