[proofplan] We rewrite the angular differential of $\exp_p$ as the value at time $r$ of a Jacobi field along the radial geodesic from $p$. The curvature pinching hypotheses allow Rauch comparison, with the upper curvature bound $K_M \le k_2$ giving the lower metric estimate and the lower curvature bound $K_M \ge k_1$ giving the upper metric estimate. The radial identities follow directly from the fact that radial lines in $T_pM$ exponentiate to unit-speed geodesics, together with linearity of the differential. [/proofplan]

[step:Convert the angular differential into a Jacobi field]Let $e := v/r \in T_pM$, so $|e|_g=1$ and $v=re$. Since $(v,w)_g=0$, we have $(e,w)_g=0$. Define $u := w/r \in T_pM$, so $(e,u)_g=0$ and $w=ru$. Define the radial geodesic \begin{align*} \gamma:[0,r] &\to M,\\ t &\mapsto \exp_p(te). \end{align*} Since $|e|_g=1$, $\gamma$ is parametrized by arclength and $\gamma([0,r]) \subset B(p,R)$. Define the geodesic variation \begin{align*} F:(-\varepsilon,\varepsilon)\times [0,r] &\to M,\\ (s,t) &\mapsto \exp_p(t(e+s u)), \end{align*} for $\varepsilon>0$ sufficiently small that $t(e+s u)$ lies in the domain of $\exp_p$ for all $(s,t) \in (-\varepsilon,\varepsilon)\times[0,r]$. Let \begin{align*} J:[0,r] &\to TM,\\ t &\mapsto \left.\frac{\partial F}{\partial s}\right|_{s=0}(t). \end{align*} Then $J$ is the variational vector field of a variation through geodesics, hence a Jacobi field along $\gamma$ with \begin{align*} J(0)=0, \qquad D_tJ(0)=u. \end{align*} At $t=r$, differentiating the map $s \mapsto \exp_p(r(e+s u))$ at $s=0$ gives \begin{align*} J(r) = (d\exp_p)_{re}(ru) = (d\exp_p)_v(w). \end{align*}[/step]

[guided]The differential of the exponential map is most naturally studied by varying the initial velocity of a geodesic. Since $v \ne 0$, define \begin{align*} e := \frac{v}{r}, \end{align*} where $r=|v|_g$. Then $|e|_g=1$ and $v=re$. Because $w$ is orthogonal to $v$, it is also orthogonal to $e$. We scale $w$ by $r$ and define \begin{align*} u := \frac{w}{r}. \end{align*} Thus $w=ru$ and $(e,u)_g=0$. Now define the radial geodesic \begin{align*} \gamma:[0,r] &\to M,\\ t &\mapsto \exp_p(te). \end{align*} Completeness ensures that $\exp_p$ is defined on all of $T_pM$, and $|e|_g=1$ ensures that $\gamma$ has unit speed. Since $0 \le t \le r < R$, every point $\gamma(t)$ lies in $B(p,R)$, so the curvature bounds from the theorem apply along this geodesic. To connect $(d\exp_p)_{re}(ru)$ with a Jacobi field, vary the initial velocity $e$ in the transverse direction $u$. For sufficiently small $\varepsilon>0$, define \begin{align*} F:(-\varepsilon,\varepsilon)\times[0,r] &\to M,\\ (s,t) &\mapsto \exp_p(t(e+s u)). \end{align*} For each fixed $s$, the curve $t \mapsto F(s,t)$ is the geodesic starting at $p$ with initial velocity $e+s u$. Therefore the variational vector field \begin{align*} J:[0,r] &\to TM,\\ t &\mapsto \left.\frac{\partial F}{\partial s}\right|_{s=0}(t) \end{align*} is a Jacobi field along $\gamma$ (citing a result not yet in the wiki: geodesic variation characterization of Jacobi fields). The initial conditions come from differentiating the variation at $t=0$. Since $F(s,0)=p$ for every $s$, \begin{align*} J(0)=0. \end{align*} The initial covariant derivative records the derivative of the initial velocity: \begin{align*} D_tJ(0) = \left.\frac{D}{ds}\right|_{s=0}\left.\frac{\partial F}{\partial t}\right|_{t=0} = \left.\frac{D}{ds}\right|_{s=0}(e+s u) = u. \end{align*} Finally, at time $r$, the curve in $M$ obtained by varying $s$ is \begin{align*} s \mapsto F(s,r)=\exp_p(r(e+s u)). \end{align*} Its derivative at $s=0$ is exactly the differential of $\exp_p$ at $re$ applied to $ru$: \begin{align*} J(r) = (d\exp_p)_{re}(ru). \end{align*} Since $re=v$ and $ru=w$, this is also \begin{align*} J(r)=(d\exp_p)_v(w). \end{align*}[/guided]

[step:Apply Rauch comparison to bound the Jacobi field] The Jacobi field $J$ satisfies $J(0)=0$ and $D_tJ(0)=u$, with $u \perp e=\dot{\gamma}(0)$. The hypothesis on sectional curvature along $\gamma([0,r]) \subset B(p,R)$ gives \begin{align*} k_1 \le K_M(\dot{\gamma}(t),\Pi_t) \le k_2 \end{align*} for every $t \in [0,r]$ and every two-plane $\Pi_t \subset T_{\gamma(t)}M$ containing $\dot{\gamma}(t)$. Since $\exp_p$ is nonsingular at $re$ and $r$ lies before the first positive zero of $\operatorname{sn}_{k_2}$, Rauch comparison applies to $J$ and to the constant-curvature model Jacobi fields with initial derivative of length $|u|_g$ (citing a result not yet in the wiki: Rauch Comparison Theorem). Therefore \begin{align*} \operatorname{sn}_{k_2}(r)|u|_g \le |J(r)|_g \le \operatorname{sn}_{k_1}(r)|u|_g. \end{align*} Using $J(r)=(d\exp_p)_{re}(ru)$ gives the equivalent angular estimate \begin{align*} \operatorname{sn}_{k_2}(r)|u|_g \le |(d\exp_p)_{re}(ru)|_g \le \operatorname{sn}_{k_1}(r)|u|_g. \end{align*} Since $u=w/r$, this becomes \begin{align*} \frac{\operatorname{sn}_{k_2}(r)}{r}|w|_g \le |(d\exp_p)_v(w)|_g \le \frac{\operatorname{sn}_{k_1}(r)}{r}|w|_g. \end{align*} [/step]

[step:Compute the radial part of the differential] Let $e \in T_pM$ be a unit vector and define \begin{align*} \gamma_e:[0,R) &\to M,\\ t &\mapsto \exp_p(te). \end{align*} Then $\gamma_e$ is the unit-speed geodesic with initial velocity $e$. For fixed $r \in (0,R)$, the curve \begin{align*} \alpha:(-\delta,\delta) &\to M,\\ s &\mapsto \exp_p((r+s)e) \end{align*} is equal to $s \mapsto \gamma_e(r+s)$ for sufficiently small $\delta>0$. Hence \begin{align*} (d\exp_p)_{re}(e) = \alpha'(0) = \dot{\gamma}_e(r). \end{align*} Because $\gamma_e$ is unit speed, \begin{align*} |(d\exp_p)_{re}(e)|_g = |\dot{\gamma}_e(r)|_g = 1. \end{align*} By linearity of the differential $(d\exp_p)_{re}:T_{re}(T_pM)\cong T_pM \to T_{\exp_p(re)}M$, \begin{align*} (d\exp_p)_{re}(re) = r(d\exp_p)_{re}(e). \end{align*} Taking norms gives \begin{align*} |(d\exp_p)_{re}(re)|_g = r|(d\exp_p)_{re}(e)|_g = r. \end{align*} This proves the stated radial identities and completes the metric comparison in [normal coordinates](/theorems/2713). [/step]