[proofplan] We compare the given Jacobi field with the radial Jacobi field in the simply [connected space](/page/Connected%20Space) form of constant sectional curvature $k$. The model field has the same initial speed and its length is exactly $|D_tJ(0)|\operatorname{sn}_k(t)$. The lower-curvature half of the Rauch comparison theorem gives that, before the first conjugate point, the length of the Jacobi field in $M$ is bounded above by the model length. The no-conjugate-point hypothesis is precisely the interval on which this comparison is valid. [/proofplan]

[step:Choose the constant-curvature model Jacobi field with matching initial speed] Let $v := D_tJ(0) \in T_{\gamma(0)}M$. Since $J$ is transverse and $J(0)=0$, the Jacobi equation implies $v \perp \dot{\gamma}(0)$. If $v=0$, then the uniqueness theorem for the linear Jacobi equation gives $J \equiv 0$ on $[0,b)$, and the desired estimate follows. Hence assume $v \neq 0$. Let $(\widetilde{M},\widetilde{g})$ be the simply connected complete Riemannian manifold of constant sectional curvature $k$. Choose a unit-speed geodesic \begin{align*} \widetilde{\gamma}:[0,\pi/\sqrt{k}) \to \widetilde{M} \end{align*} and a vector $\widetilde{v} \in T_{\widetilde{\gamma}(0)}\widetilde{M}$ such that $\widetilde{v} \perp \dot{\widetilde{\gamma}}(0)$ and $|\widetilde{v}|_{\widetilde{g}}=|v|_g$. Let \begin{align*} \widetilde{P}_s:T_{\widetilde{\gamma}(0)}\widetilde{M} \to T_{\widetilde{\gamma}(s)}\widetilde{M} \end{align*} denote parallel transport along $\widetilde{\gamma}$ from time $0$ to time $s$. Let $D_s$ denote covariant differentiation along $\widetilde{\gamma}$ with respect to the Levi-Civita connection of $\widetilde{g}$. Define the model Jacobi field \begin{align*} \widetilde{J}:[0,\pi/\sqrt{k}) &\to T\widetilde{M} \\ s &\mapsto \operatorname{sn}_k(s)\,\widetilde{P}_s\widetilde{v}. \end{align*} Because $\widetilde{P}_s\widetilde{v}$ is parallel and orthogonal to $\dot{\widetilde{\gamma}}(s)$, the constant-curvature Jacobi equation becomes \begin{align*} D_s^2\widetilde{J}+k\widetilde{J}=0. \end{align*} The scalar function $\operatorname{sn}_k$ satisfies $\operatorname{sn}_k(0)=0$ and $\operatorname{sn}_k'(0)=1$, so $\widetilde{J}(0)=0$ and $D_s\widetilde{J}(0)=\widetilde{v}$. Therefore \begin{align*} |\widetilde{J}(s)|_{\widetilde{g}}=|\widetilde{v}|_{\widetilde{g}}\,\operatorname{sn}_k(s)=|D_tJ(0)|_g\,\operatorname{sn}_k(s) \end{align*} for every $s \in (0,\pi/\sqrt{k})$. [/step]

[step:Apply Rauch comparison before the first conjugate point]Fix $t \in (0,\min\{b,\pi/\sqrt{k}\})$ and assume that $\gamma(0)$ has no conjugate point to $\gamma(s)$ along $\gamma$ for $s \in (0,t)$. Let $K_{\widetilde{g}}$ denote the sectional curvature function of the model manifold $(\widetilde{M},\widetilde{g})$. For each $t' \in (0,t)$, we apply the lower-curvature half of the Rauch comparison theorem on the interval $(0,t']$ to the Jacobi fields $J$ and $\widetilde{J}$. The hypotheses of Rauch comparison are satisfied as follows. Both $\gamma$ and $\widetilde{\gamma}$ are unit-speed geodesics. The fields $J$ and $\widetilde{J}$ are transverse Jacobi fields with \begin{align*} J(0)=0,\qquad \widetilde{J}(0)=0,\qquad |D_tJ(0)|_g=|D_s\widetilde{J}(0)|_{\widetilde{g}}. \end{align*} For each $s \in (0,t)$, let $\sigma_s \subset T_{\gamma(s)}M$ denote any $2$-plane containing $\dot{\gamma}(s)$, and let $\widetilde{\sigma}_s \subset T_{\widetilde{\gamma}(s)}\widetilde{M}$ denote any $2$-plane containing $\dot{\widetilde{\gamma}}(s)$. Every sectional curvature of $\widetilde{M}$ along such a plane $\widetilde{\sigma}_s$ is equal to $k$, while the assumed curvature bound gives \begin{align*} K_g(\sigma_s) \geq k = K_{\widetilde{g}}(\widetilde{\sigma}_s) \end{align*} for these comparison planes. Finally, the no-conjugate-point assumption on $(0,t)$ is exactly the nonsingularity condition required for the comparison on every compact subinterval $[0,t']$ with $0<t'<t$. Rauch comparison therefore yields, for every $t' \in (0,t)$, \begin{align*} |J(t')|_g \leq |\widetilde{J}(t')|_{\widetilde{g}}. \end{align*} Using the explicit length of the model field computed above gives \begin{align*} |J(t')|_g \leq |D_tJ(0)|_g\,\operatorname{sn}_k(t'). \end{align*} The maps $s \mapsto |J(s)|_g$ and $s \mapsto \operatorname{sn}_k(s)$ are continuous on $[0,t]$, because Jacobi fields are smooth along geodesics and $\operatorname{sn}_k$ is smooth. Taking the limit $t' \uparrow t$ gives \begin{align*} |J(t)|_g \leq |D_tJ(0)|_g\,\operatorname{sn}_k(t). \end{align*}[/step]

[guided]The purpose of the comparison field is to isolate the effect of the curvature lower bound. Let $v := D_tJ(0) \in T_{\gamma(0)}M$. Since $J(0)=0$ and $J$ is transverse, we have $v \perp \dot{\gamma}(0)$. Choose the simply connected complete Riemannian manifold $(\widetilde{M},\widetilde{g})$ of constant sectional curvature $k$, choose a unit-speed geodesic \begin{align*} \widetilde{\gamma}:[0,\pi/\sqrt{k}) \to \widetilde{M}, \end{align*} and choose $\widetilde{v} \in T_{\widetilde{\gamma}(0)}\widetilde{M}$ with $\widetilde{v} \perp \dot{\widetilde{\gamma}}(0)$ and $|\widetilde{v}|_{\widetilde{g}}=|v|_g$. For $s \in [0,\pi/\sqrt{k})$, let \begin{align*} \widetilde{P}_s:T_{\widetilde{\gamma}(0)}\widetilde{M} \to T_{\widetilde{\gamma}(s)}\widetilde{M} \end{align*} denote parallel transport along $\widetilde{\gamma}$ from time $0$ to time $s$, and define \begin{align*} \widetilde{J}:[0,\pi/\sqrt{k}) &\to T\widetilde{M} \\ s &\mapsto \operatorname{sn}_k(s)\,\widetilde{P}_s\widetilde{v}. \end{align*} Because $\widetilde{P}_s\widetilde{v}$ is parallel and remains orthogonal to $\dot{\widetilde{\gamma}}(s)$, the constant-curvature Jacobi equation reduces to $D_s^2\widetilde{J}+k\widetilde{J}=0$. Also $\operatorname{sn}_k(0)=0$ and $\operatorname{sn}_k'(0)=1$, so $\widetilde{J}(0)=0$ and $D_s\widetilde{J}(0)=\widetilde{v}$. Since parallel transport preserves length, \begin{align*} |\widetilde{J}(s)|_{\widetilde{g}} = \operatorname{sn}_k(s)|\widetilde{v}|_{\widetilde{g}} = |D_tJ(0)|_g\,\operatorname{sn}_k(s). \end{align*} Now fix $t' \in (0,t)$. We invoke the lower-curvature half of Rauch comparison on $[0,t']$. This theorem compares transverse Jacobi fields whose initial values vanish and whose initial speeds have the same length. Its curvature hypothesis is that the sectional curvatures of the first manifold along planes containing the geodesic velocity are bounded below by the sectional curvatures of the comparison manifold. Let $K_{\widetilde{g}}$ denote the sectional curvature function of $(\widetilde{M},\widetilde{g})$. Here the curvature condition is exactly \begin{align*} K_g(\sigma) \geq k = K_{\widetilde{g}}(\widetilde{\sigma}) \end{align*} for every plane $\sigma \subset T_{\gamma(s)}M$ containing $\dot{\gamma}(s)$ and every plane $\widetilde{\sigma} \subset T_{\widetilde{\gamma}(s)}\widetilde{M}$ containing $\dot{\widetilde{\gamma}}(s)$, with $0<s<t'$. The theorem also requires that the comparison be made before the first conjugate point along $\gamma$, because conjugate points are precisely where the differential of the exponential map can become singular and the length-ratio comparison may fail. This is why we apply the theorem first on $[0,t']$, where $t'<t$. Rauch comparison gives \begin{align*} |J(t')|_g \leq |\widetilde{J}(t')|_{\widetilde{g}}. \end{align*} Substituting the explicit model length at time $t'$ yields \begin{align*} |J(t')|_g \leq |D_tJ(0)|_g\,\operatorname{sn}_k(t'). \end{align*} Finally, $s \mapsto |J(s)|_g$ is continuous because $J$ is a smooth Jacobi field, and $s \mapsto \operatorname{sn}_k(s)$ is continuous because it is smooth. Letting $t' \uparrow t$ gives \begin{align*} |J(t)|_g \leq |D_tJ(0)|_g\,\operatorname{sn}_k(t). \end{align*} This is the desired infinitesimal upper bound on transverse separation before conjugacy.[/guided]

[step:Conclude the estimate on the stated interval] Since $t \in (0,\min\{b,\pi/\sqrt{k}\})$ was arbitrary subject to the absence of conjugate points on $(0,t)$, the estimate holds for every such $t$. The expression \begin{align*} \operatorname{sn}_k(t)=\frac{\sin(\sqrt{k}\,t)}{\sqrt{k}} \end{align*} is positive on $(0,\pi/\sqrt{k})$, so the comparison field is nonzero on the whole interval before its first zero at $\pi/\sqrt{k}$. This completes the proof. [/step]