Existence uniqueness and smooth dependence of geodesics

Apply [F2] to the spray at the point $v\in TM$. It gives a unique maximal integral curve $Z_v:I_v\to TM$ on an open interval containing zero. By [F1], $Z_v$ is the velocity lift of ${\gamma }_{p,v}:=\pi \circ Z_v$. Since $Z_v(0)=v$, its base point is $p$, and the base component of the spray equation gives ${\gamma }_{p,v}^{\mathrm{\prime }}(0)=v$.

If a geodesic with these initial data existed on a larger interval, [F1] would make its velocity lift an integral curve of the spray extending $Z_v$, contrary to maximality. The same lift argument and integral-curve uniqueness prove uniqueness on every common interval. Thus $I_{p,v}=I_v$ and the geodesic is uniquely maximal.

By [F3], $\{(t,v):t\in I_v\}$ is open in $\mathbb{R}\times TM$ and $(t,v)\mapsto Z_v(t)$ is smooth. This is exactly $\mathcal{G}$ after writing $v$ together with its determined base point $p$. In induced tangent-bundle coordinates the projection $\pi (x,w)=x$ is smooth, so composing gives the asserted smooth geodesic evaluation.

For $v=0$, [F1] makes $Z_v$ stationary and the maximal geodesic is the constant curve on all of $\mathbb{R}$. In dimension zero every initial vector is zero; for empty $M$ there are no initial vectors. Each maximal domain is open, so it has no included finite endpoints. All conclusions concern one supplied initial vector at a time. The only choice principle is the declared ${\mathrm{A}\mathrm{C}}_{\omega }$, inherited exactly from the smooth-manifold structure on $TM$; [F2]–[F3] then apply without another family selection.