Every Jacobi field is induced by a geodesic variation

Put $L=b-a$, $q=\gamma (a)$, $v_0=\dot{\gamma }(a)$, $A=J(a)$, and $B=D_{t}J(a)$. By [F3], choose a smooth curve $\sigma :(-\delta ,\delta )\to M$ with $\sigma (0)=q$ and ${\sigma }^{\mathrm{\prime }}(0)=A$. By [F4], let $X$ and $W$ be the unique parallel sections along $\sigma$ with $X(0)=v_0$ and $W(0)=B$. Their values are the transports $X(s)=P_{\sigma ;0,s}{v}_0$ and $W(s)=P_{\sigma ;0,s}B$ by [F6]. By [F5], $D_{s}X=D_{s}W=0$. Define $u(s)=X(s)+sW(s)\in T_{\sigma (s)}M$. In a local bundle chart along a short subinterval of $\sigma$, the coefficient functions of $X$ and $W$ are smooth, so those of $u$ are smooth; [F7] therefore makes $u$ a smooth curve in $TM$. Also $u(0)=v_0$.

By uniqueness of the initial-value geodesic, $r\mapsto \gamma (a+r)$ is ${\gamma }_{q,v_0}(r)$ for $r\in [0,L]$. Thus $K=\{(r,(q,v_0)):r\in [0,L]\}$ lies in the open geodesic-flow domain $\mathcal{G}\subset \mathbb{R}\times TM$ from [F2]. The map $H(s,r)=(r,(\sigma (s),u(s)))$ into $\mathbb{R}\times TM$ is smooth by [F8] and $H(0,r)\in K$ for every $r$. The open set $H^{-1}(\mathcal{G})$ contains $\{0\}\times [0,L]$, so for each $r_0\in [0,L]$ it contains a product neighborhood $(-{\epsilon }_{r_0},{\epsilon }_{r_0})\times (r_0-{\eta }_{r_0},r_0+{\eta }_{r_0})$. Compactness gives finitely many of these second intervals covering $[0,L]$; the minimum of their positive first radii is one $\epsilon >0$ with $(-\epsilon ,\epsilon )\times [0,L]\subset H^{-1}(\mathcal{G})$. Therefore $\tilde{F}(s,r)=G(r,\sigma (s),u(s))$ is smooth by [F8], each $r$-curve is an affinely parametrized geodesic by [F9], and $\tilde{F}(0,r)=\gamma (a+r)$. Set $F(s,t)=\tilde{F}(s,t-a)$. Then $F$ is smooth on the common rectangle, has central curve $\gamma$, and is a geodesic variation by [F10].

Let $V(t)={\mathrm{\partial }}_{s}F(0,t)$. By [F11], $V$ is Jacobi. Since $F(s,a)=\sigma (s)$, $V(a)=A=J(a)$. In a chart about $q$, write $\tilde{F}$ in coordinates $x^k(s,r)$. By the initial-velocity clause in [F2], ${\mathrm{\partial }}_r\tilde{F}(s,0)=u(s)$. The components of $D_r({\mathrm{\partial }}_s\tilde{F})$ and $D_s({\mathrm{\partial }}_r\tilde{F})$ are ${\mathrm{\partial }}_r{\mathrm{\partial }}_s{x}^k+{\mathrm{\Gamma }}_{ij}^k{\mathrm{\partial }}_r{x}^i{\mathrm{\partial }}_s{x}^j$ and ${\mathrm{\partial }}_s{\mathrm{\partial }}_r{x}^k+{\mathrm{\Gamma }}_{ij}^k{\mathrm{\partial }}_s{x}^i{\mathrm{\partial }}_r{x}^j$. Coordinate vector fields commute and torsion freeness gives ${\mathrm{\Gamma }}_{ij}^k={\mathrm{\Gamma }}_{ji}^k$, so these components agree and $D_r({\mathrm{\partial }}_s\tilde{F})(0,0)=D_{s}u(0)$. By [F5] and the product rule in [F13], $D_{s}u(0)=D_s(X+sW)(0)=W(0)=B$. Hence $D_{t}V(a)=B=D_{t}J(a)$, with the one-sided derivative at the included endpoint.

Both $V$ and $J$ are Jacobi fields along the same segment with the same initial data, so [F12] gives $V=J$ on all of $[a,b]$ and the constructed $F$ has the prescribed variation field. If $M$ is empty there is no supplied geodesic segment. In dimension zero, $A=B=0$, the unique parallel sections and $u$ are zero, and the construction is the constant variation; dimension one is covered by the same argument. If $J=0$, then $A=B=0$ and the construction still works. If $\gamma$ is constant, then $v_0=0$ and the construction varies its initial point and velocity while matching both Jacobi initial data. Endpoints are included with one-sided derivatives and may move. The only choice assumption is exactly ${\mathrm{A}\mathrm{C}}_{\omega }$ through [F1]; the geodesic-flow supplier [F2] carries the same premise. The parallel extensions are unique and the common interval uses only a finite cover of $[0,L]$, so no full Axiom of Choice is used. This is a one-way existence assertion, not an iff claim.