Local isometries send geodesics to geodesics

Fix $p\in M$. By [F1], there are open sets $p\in U\subseteq M$ and $F(p)\in V\subseteq N$ for which $f=F{\mathrm{\mid }}_U:U\to V$ is a diffeomorphism. Shrinking $U$ to a coordinate neighborhood of $p$ and replacing $V$ by its image preserves this property. The pullback identity restricts to $f^{\ast }(h{\mathrm{\mid }}_V)=g{\mathrm{\mid }}_U$, so $f$ is an isometry between these neighborhoods.

For vector fields $X,Y$ on $U$, define ${\bar{\mathrm{\nabla }}}_{X}Y=(f^{-1}{)}_{\ast }({\mathrm{\nabla }}_{f_{\ast }X}^N(f_{\ast }Y))\mathrm{.}$ This is well defined because a diffeomorphism sends vector fields on $U$ bijectively to vector fields on $V$. For $a\in C^{\mathrm{\infty }}(U)$ one has $f_{\ast }(aX)=(a\circ f^{-1})f_{\ast }X$, $f_{\ast }(aY)=(a\circ f^{-1})f_{\ast }Y$, and $(f_{\ast }X)(a\circ f^{-1})=X(a)\circ f^{-1}$. The connection axioms for ${\mathrm{\nabla }}^N$ therefore give $C^{\mathrm{\infty }}(U)$-linearity in $X$, real-linearity in $Y$, and ${\bar{\mathrm{\nabla }}}_X(aY)=X(a)Y+a{\bar{\mathrm{\nabla }}}_{X}Y$. Thus $\bar{\mathrm{\nabla }}$ is an affine connection on $U$.

Diffeomorphisms preserve brackets: for every $u\in C^{\mathrm{\infty }}(V)$, $[f_{\ast }X,f_{\ast }Y](u)=([X,Y](u\circ f))\circ f^{-1},$ so $[f_{\ast }X,f_{\ast }Y]=f_{\ast }[X,Y]$. Since ${\mathrm{\nabla }}^N$ is torsion free, $f_{\ast }({\bar{\mathrm{\nabla }}}_{X}Y-{\bar{\mathrm{\nabla }}}_{Y}X-[X,Y])={\mathrm{\nabla }}_{f_{\ast }X}^N(f_{\ast }Y)-{\mathrm{\nabla }}_{f_{\ast }Y}^N(f_{\ast }X)-[f_{\ast }X,f_{\ast }Y]=0.$ The differential of $f$ is invertible, hence $\bar{\mathrm{\nabla }}$ is torsion free.

Because $g(Y,Z)=h(f_{\ast }Y,f_{\ast }Z)\circ f$, the chain rule and metric compatibility of ${\mathrm{\nabla }}^N$ give $X(g(Y,Z))=g({\bar{\mathrm{\nabla }}}_{X}Y,Z)+g(Y,{\bar{\mathrm{\nabla }}}_{X}Z)\mathrm{.}$ Indeed, after composing this scalar identity with $f^{-1}$ and using step 2.1, it is exactly $(f_{\ast }X)h(f_{\ast }Y,f_{\ast }Z)=h({\mathrm{\nabla }}_{f_{\ast }X}^N{f}_{\ast }Y,f_{\ast }Z)+h(f_{\ast }Y,{\mathrm{\nabla }}_{f_{\ast }X}^N{f}_{\ast }Z)\mathrm{.}$ Thus $\bar{\mathrm{\nabla }}$ is compatible with $g{\mathrm{\mid }}_U$.

The connection $\bar{\mathrm{\nabla }}$ and the restriction of ${\mathrm{\nabla }}^M$ to $U$ are both torsion free and compatible with $g{\mathrm{\mid }}_U$. Uniqueness in [F2] gives $\bar{\mathrm{\nabla }}={\mathrm{\nabla }}^M{\mathrm{\mid }}_U$. Applying $f_{\ast }$ to the definition in step 2.1 yields the asserted local intertwining identity.

Fix $t_0\in I$ and use step 1.1 at $\gamma (t_0)$. On a relative interval $J\subseteq I$ about $t_0$ with $\gamma (J)\subseteq U$, take the coordinate frame $E_1,\dots ,E_n$ on $U$ and write $W(t)=w^i(t)E_i{\mathrm{\mid }}_{\gamma (t)}$. The defining coefficient rule for covariant differentiation along a curve and step 4.1 give $\begin{array}{rl}{\displaystyle D_t^N(dF(W))}& {\displaystyle =(w^i{)}^{\mathrm{\prime }}{f}_{\ast }{E}_i+w^i{\mathrm{\nabla }}_{(f\circ \gamma {)}^{\mathrm{\prime }}}^N(f_{\ast }{E}_i)}\\ {\displaystyle }& {\displaystyle =df((w^i{)}^{\mathrm{\prime }}{E}_i+w^i{\mathrm{\nabla }}_{{\gamma }^{\mathrm{\prime }}}^M{E}_i)=df(D_t^{M}W)\mathrm{.}}\end{array}$ on $J$. Since $t_0$ was arbitrary, the identity holds on all of $I$.

Apply step 5.1 to $W={\gamma }^{\mathrm{\prime }}$. Then $dF({\gamma }^{\mathrm{\prime }})=(F\circ \gamma {)}^{\mathrm{\prime }}$, and if $\gamma$ is geodesic, [F3] gives $D_t^N(F\circ \gamma {)}^{\mathrm{\prime }}=dF(D_t^M{\gamma }^{\mathrm{\prime }})=dF(0)=0.$ Therefore $F\circ \gamma$ is an affinely parametrized geodesic. This uses only the local connection identity, so none of completeness, connectedness, surjectivity, or global injectivity enters.

Constant curves are covered because their velocity and acceleration are zero. If $M$ is empty there are no curves to check; in dimension zero every curve from an interval is locally constant, and the same conclusion holds. The proof is unchanged in dimension one. At an included endpoint, step 5.1 is read on a one-sided relative interval and [F3] supplies the one-sided covariant derivative. Each neighborhood is used only after fixing one supplied point or parameter value; no simultaneous selection of neighborhoods is made, so the argument uses no form of the Axiom of Choice. The claim is one-way rather than an equivalence, and it concerns affine parametrization, not preservation of minimizing behavior.