How Do You Calculate Curvature from a Given Metric?

[stephenkeiths]

Homework Statement

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]

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!

 

[mighty2000]

Hello,

Thanks for posting your attempt at solving this problem. From what I can see, you have followed the correct steps in using Cartan's method to find the curvature of the given metric. However, I believe you may have made a computational error when plugging in the values for A and B.

When A=B=1/v^2, the curvature should actually be -4/v^4, not -2v^2. This can be seen by plugging in the values for A and B into the expression for K that you found:

K = (1/AB)[(1/A)(d^2B/du^2) - (1/B)(d^2A/dv^2) - (1/A^2)(dA/du)(dB/du) + (1/B^2)(dA/dv)(dB/dv)]

= (1/(1/v^4))[(1/(1/v^2))(d^2(1/v^2)/du^2) - (1/(1/v^2))(d^2(1/v^2)/dv^2) - (1/(1/v^4))(d(1/v^2)/du)(d(1/v^2)/du) + (1/(1/v^4))(d(1/v^2)/dv)(d(1/v^2)/dv)]

= (v^4)[(-2/v^6) - (-2/v^6) - (-4/v^8) + (-4/v^8)]

= (-4/v^4)

Therefore, the curvature for the Poincare metric should be -4/v^4, which is equivalent to -1 when v=1 (as it does in the Poincare metric).

I hope this helps. Keep up the good work!