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Let $(M,g)$ be a riemannian manifold with Levi-Civita connection $\nabla$. A geodesic $c:[0,1]\rightarrow M$ is solution of the following equation:

$$2 (\ddot{c})^* =\nabla (g) (\dot{c},\dot{c})$$

Can we integrate geodesics over riemannian manifolds?

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