# Computing Curvature in 2D

1. May 1, 2013

### stephenkeiths

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

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

Then I use Cartan's second structural equation to find the curvature K (which is just the coefficient of $dw$). I find
$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}$$\frac{dB}{dv}]$

But then when I plug in $A=B=\frac{1}{v^{2}}$ I get $-2v^{2}$

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