Differential of the exponential map in terms of Jacobi fields

Since ${\mathcal{E}}_p$ is open and contains $v$, choose $\epsilon >0$ such that $v+sw\in {\mathcal{E}}_p$ for every $\mathrm{\mid }s\mathrm{\mid }<\epsilon$; if $w=0$ any positive $\epsilon$ works. For each such $s$ and every $t\in [0,1]$, [F2] gives $t(v+sw)\in {\mathcal{E}}_p$ and ${\exp }_p(t(v+sw))={\gamma }_{p,v+sw}(t)$. The map $(s,t)\mapsto t(v+sw)$ is smooth into ${\mathcal{E}}_p$, so [F1] and [F8] make $F(s,t):={\exp }_p(t(v+sw))$ smooth on the common rectangle $(-\epsilon ,\epsilon )\times [0,1]$. Each longitudinal curve is the affinely parametrized geodesic ${\gamma }_{p,v+sw}$ restricted to $[0,1]$, so [F3] makes $F$ a geodesic variation with central curve $\gamma$.

Put $J(t)={\mathrm{\partial }}_{s}F(0,t)$. By [F4], $J$ is Jacobi, and $F(s,0)=p$ gives $J(0)=0$. For fixed $t\in [0,1]$, the curve $c_t(s)=t(v+sw)$ lies in ${\mathcal{E}}_p$ and has $c_t(0)=tv$, ${\dot{c}}_t(0)=tw$. Applying [F6] to ${\exp }_p\circ c_t$ yields $J(t)={\frac{d}{ds}\mid }_{s=0}{\exp }_p(c_t(s))=d({\exp }_p{)}_{tv}(tw)$. Here $T_{tv}({\mathcal{E}}_p)\cong T_{p}M$ canonically because ${\mathcal{E}}_p$ is open in the vector space $T_{p}M$. At $t=0$, $c_0$ is constant and both sides vanish.

To compute the initial covariant derivative, take a chart about $p$ and write $F^i$ for its coordinate functions near $(0,0)$ and $J^i(t)={\mathrm{\partial }}_s{F}^i(0,t)$. By [F7], $(D_{t}J{)}^i=(J^i{)}^{\mathrm{\prime }}+{\mathrm{\Gamma }}_{jk}^i{\dot{\gamma }}^j{J}^k$; since $F(s,0)=p$, $J(0)=0$, so the connection term at $0$ vanishes. The mixed partial derivatives commute, and [F2] gives ${\mathrm{\partial }}_t{F}^i(s,0)=v^i+s{w}^i$ in the fixed tangent space $T_{p}M$. Therefore $(D_{t}J{)}^i(0)={\mathrm{\partial }}_t{\mathrm{\partial }}_s{F}^i(0,0)={\mathrm{\partial }}_s{\mathrm{\partial }}_t{F}^i(0,0)=w^i$, including the one-sided derivative, so $D_{t}J(0)=w$.

By [F5], $J$ is the unique Jacobi field with initial data $(0,w)$, so step 2.1 proves the formula for every $t$, and at $t=1$ gives $d({\exp }_p{)}_v(w)=J(1)$. If $M$ is empty there is no supplied $p$; in dimension zero $v=w=0$ and the unique field and both sides are zero; in dimension one the argument is unchanged. If $v=0$, the central geodesic is constant and the same construction applies; if $w=0$, the variation is constant in $s$ and both sides are zero. The endpoints $t=0,1$ use one-sided derivatives. The only choice assumption is exactly ${\mathrm{A}\mathrm{C}}_{\omega }$, inherited through the exponential domain and smooth geodesic-flow suppliers [F1, F2]; selecting one neighborhood size for the supplied $v,w$ uses openness and no choice function, and no full Axiom of Choice is used. This is a one-way formula, not an iff claim.