Geodesic of an affine connection

The velocity ${\gamma }^{\mathrm{\prime }}$ is a section of ${\gamma }^{\ast }TM$, so [F2] makes $D_t{\gamma }^{\mathrm{\prime }}$ well defined and intrinsic. The equation therefore compares vectors in $T_{\gamma (t)}M$ and is independent of any chart or extension of the velocity field.

If $\gamma (t)=p$ is constant, then ${\gamma }^{\mathrm{\prime }}$ is the zero section and [F2] gives $D_t{\gamma }^{\mathrm{\prime }}=0$, so constant curves are included. On a zero-dimensional manifold every smooth curve on an interval is locally constant and hence has zero velocity; the empty manifold has no such curves. A singleton parameter interval is excluded because [F2] supplies no derivative operator there. Included interval endpoints use the one-sided convention, and no point, chart, or curve is selected from a family, so no choice principle is used.