= Solution
The metric $g$ is <geodesic completeness>[geodesically complete] when every maximal affinely parametrized <geodesic> is defined on all of $\mathbb R$; equivalently, $\exp_p$ is defined on every $T_pM$ for every $p\in M$.
The <Hopf-Rinow theorem> says that for a connected <Riemannian manifold>, the following are equivalent: geodesic completeness; completeness of the <Riemannian distance> $d_g$; compactness of every closed bounded subset; and the existence, between every two points, of a length-minimizing geodesic. It is enough in the exponential-map formulation that $\exp_p$ be defined on all of $T_pM$ for one point $p$.
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