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I'd like to prove that the hyperbolic plane is complete in a short, nice way. Here's my proof, but I'm not convinced of its correctness yet. Let $\mathbb{H}=\left\{(x,y)\in\mathbb{R}^2\big\vert y>0\right\}$. Since this is a connected, Riemannian manifold with infinitesimal metric $dh^2=ds^2/y^2)$ (where $ds$ is the usual Euclidean metric) we can use the Hopf-Rinow theorem and say that completeness is equivalent to having geodesics defined from the whole real line. Hence, it suffices to show that every point $p\in\mathbb{H}$ is at infinite distance from the boundary. Take $q\in\partial\mathbb{H}$. By moving the points via isometries, we can assume that $p$ lies in the positive, pure imaginary, open semi-line and have coordinates $p=(0,\bar{y})$ and that $q$ is the point $\infty$. Then we may finish the proof just computing the integral $$\int_{\bar{y}}^\infty \sqrt{\frac{dy^2}{y^2}}=log(|y|)\Big\vert_{\bar{y}}^\infty=\infty.$$ Is there any wrong step? Do you know any other concise proof of the completeness of the hyperbolic plane?

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If you're okay using Hopf-Rinow, there's a cute 2-step proof: (1) show that there is one geodesic that can be extended to have domain $\mathbb R$ (2) since the isometry group acts transitively on geodesics, all geodesics can be infinitely extended. – Ryan Budney Feb 9 '12 at 20:52

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