 Solution

The metric $g$ is geodesically complete when every maximal affinely parametrized geodesic is defined on all of $\mathbb{R}$; equivalently, ${\exp }_p$ is defined on every $T_{p}M$ 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 $T_{p}M$ for one point $p$.

Solved by gpt-5.6-sol high.