Local Gauss-Bonnet for a rectangle

I'm trying to satisfy the local Gauss-Bonnet equation for the surface $$R$$ bounded by $$u=A,u=B,v=a,v=b$$ in the hyperbolic plane. However I am stuck.

I know that the equation of local Gauss-Bonnet is $$\int_{\partial R} \kappa_gds +\int \int _R KdA + \sum \epsilon_j = 2\pi$$ where $$\epsilon$$ re presents the interior angles.

Now I know my shape is a square and in the hyperbolic plane hence $$dA=\sqrt{EG-F}= \frac{1}{v^2}$$ and $$K=-1$$. \

So $$\int \int _R KdA = -\int_a^b\int_A^B\frac{1}{v^2}dudv = (B-A)(\frac{1}{b}-\frac{1}{a})$$.

And $$\kappa_g$$ for the lines $$v=a,v=b$$ is $$1$$ and $$-1$$ respectively by direction. Therefore we would have that $$\int_{\partial R} \kappa_gds = \int_A^Bds-\int_A^Bds =0$$.

However then $$\sum \epsilon_j= 4\frac{\pi}{2}=2\pi$$. However this doesnt satisfy local Gauss-Bonnet since $$\int_{\partial R} \kappa_gds +\int \int _R KdA + \sum \epsilon_j = (B-A)(\frac{1}{b}-\frac{1}{a}) +2\pi$$ so I'm not sure where went wrong.

• Hmm, this question looks familiar. You have $ds$ wrong when you're doing the $\int \kappa_g\,ds$ computations. – Ted Shifrin Nov 29 '18 at 0:25