Killing fields restrict to Jacobi fields along geodesics

Fix any compact nondegenerate subinterval $[a,b]\subseteq I$ and put $K=\gamma ([a,b])$. Since $K$ is compact and the flow domain $\mathcal{D}$ is open with $\{0\}\times M\subseteq \mathcal{D}$ by [F1, F3], a finite product-neighborhood cover of $\{0\}\times K$ gives $\epsilon >0$ and an open neighborhood $U\supseteq K$ with $(-\epsilon ,\epsilon )\times U\subseteq \mathcal{D}$. Set $\mathrm{\Gamma }(s,t)=\mathrm{\Phi }(s,\gamma (t))$ on $(-\epsilon ,\epsilon )\times [a,b]$; it is smooth up to the time endpoints. This finite compactness argument makes no countable selection.

Fix $\mathrm{\mid }s\mathrm{\mid }<\epsilon$ and $p\in {\mathcal{D}}_s$, and set $q=\mathrm{\Phi }(s,p)$. The translated curve $r\mapsto \mathrm{\Phi }(r+s,p)$ is an integral curve through $q$ on the shifted interval ${\mathcal{D}}_p-s$, which contains $-s$. By maximality and uniqueness of integral curves [F3, F4], $-s\in {\mathcal{D}}_q$ and $\mathrm{\Phi }(-s,q)=p$. Reversing $s$ gives the inverse identity on ${\mathcal{D}}_{-s}$. Since both slices are smooth and their domains are open, ${\mathrm{\Phi }}_s:{\mathcal{D}}_s\to {\mathcal{D}}_{-s}$ is a diffeomorphism with inverse ${\mathrm{\Phi }}_{-s}$. Restricting to $U$ makes each ${\mathrm{\Phi }}_s$ a local diffeomorphism.

The hypothesis ${\mathcal{L}}_{X}g=0$ and [F6] give ${\mathrm{\Phi }}_s^{\ast }g=g$ on $U$ for every $\mathrm{\mid }s\mathrm{\mid }<\epsilon$. Thus each ${\mathrm{\Phi }}_s{\mathrm{\mid }}_U$ is a local Riemannian isometry by [F7].

For each fixed $s$, [F8] applied to the local isometry from step 3.1 shows that $t\mapsto \mathrm{\Phi }(s,\gamma (t))$ is an affinely parametrized geodesic. Hence $\mathrm{\Gamma }$ is a geodesic variation by [F9], with variation field $V(t)={\mathrm{\partial }}_s\mathrm{\Gamma }(0,t)$.

The integral-curve property [F2] gives $V(t)={{\mathrm{\partial }}_s\mathrm{\Phi }(s,\gamma (t))\mid }_{s=0}=X_{\mathrm{\Phi }(0,\gamma (t))}=X_{\gamma (t)}=J(t)$. The geodesic-variation theorem [F10] therefore makes $J$ Jacobi on $[a,b]$.

Since $[a,b]$ was arbitrary, the Jacobi equation holds throughout $I$, with one-sided derivatives at included endpoints. If $\gamma$ is constant, every longitudinal curve of $\mathrm{\Gamma }$ is constant in $t$, so [F10] still applies; if $X=0$, then $J=0$. The empty manifold has no supplied geodesic, dimension zero has only the zero field, and dimension one is covered by the same argument. Only finite compactness arguments are used, so neither ${\mathrm{A}\mathrm{C}}_{\omega }$ nor full AC is invoked. The proposition is one-way, not an iff claim.