For me, from the definition of a geodesic:
Let $\gamma : t \rightarrow \gamma(t)$, $t \in I$, be a curve in a manifold $M$. The curve $\gamma$ is called a geodesic if the family of tangent vectors $\dot\gamma(t)$ is parallel with respect to $\gamma$.
it is not obvious, that for any two given points in a connected manifold $M$ there is a geodesic passing through them. Is this so?