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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.

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

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