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The upper-half plane $\mathbb H$ carries a hyperbolic metric and the geodesics are semicircles with base on the real line. We consider oriented geodesics. Let $x \in \mathbb H$ and let $v$ be a unit tangent vector at $v$. How to prove the following statement:

There exists a unique geodesic $\gamma$ on $\mathbb H$ such that $\gamma^\prime(0) = v$.

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    This fact is true on any Riemannian manifold, though $\gamma$ is only neccesarily defined for small $t$ values. On a complete Riemannian manifold (like $\mathbf{H}$), $\gamma$ is defined for all time.2010-11-23

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