Boundaryless convention for geodesic flow and Hopf–Rinow

The interval gives the obstruction. Its Euclidean Christoffel symbol is zero in the interior, so an affinely parametrized geodesic with initial data $\gamma (0)=1\mathrm{/}2$ and ${\gamma }^{\mathrm{\prime }}(0)=1$ must locally be $\gamma (t)=1\mathrm{/}2+t$. It remains inside $I$ only for $-1\mathrm{/}2\le t\le 1\mathrm{/}2$ and cannot be continued as that solution for all real times while taking values in $I$.

Nevertheless [F2] makes $I$ a complete metric space. Thus the implication “metric completeness implies two-sided geodesic completeness” would be false if arbitrary manifold boundaries were silently admitted. The boundaryless convention in [F1] prevents this mismatch. The two endpoints and outward direction are explicit; the zero-dimensional case has only constant geodesics, the empty case is vacuous, and no selection or choice principle is used.