Lipschitz constant of exponential map I asked before this question on MSE but I was not able to work out the details on my own.
Suppose $M$ is a smooth compact Riemannian manifold, take $p \in M$ and consider the map
$$ T_pM \ni v \mapsto \exp_p(tv)\in M $$
where $t \in (0, \text{inj}(M))$ is a fixed parameter and $\text{inj}(M)$ is the (positive) injectivity radius of $M$.
Is is true that this map in Lipschitz uniformly in $p$? More precisely, is it possibile to prove that there exist $\delta>0$ and $C>0$ s.t. 
$$d(\exp_p(tv_1), \exp_p(tv_2)) \le Ct \|v_1-v_2\|_p $$
for every $v_1, v_2 \in T_pM$, $p \in M$ and $t \in (0, \delta)$? Clearly with $d$ I mean the Riemannian distance on $M$ and with $\| \cdot \|_p$ the norm in $T_pM$ induced by the Riemannian metric.
 A: I believe that the answer is positive. Let me consider an autonomous differential equation $\dot x=f(x)$ and let us assume that $f$ is Lipschitz-continuous. The flow $\phi(t,y)$ is defined by
$$
\dot \phi(t, y) = f\bigl(\phi(t,y)\bigr), \quad \phi(0, y)=y.
$$
As a result, we have 
$
\phi(t, y_2)-\phi(t, y_1)=y_2-y_1+\int_0^t\left(
f(\phi(s,y_2))-f(\phi(s,y_2))
\right) ds,
$
so that, at least in a coordinate chart,
$$
\rho(t)=\Vert\phi(t, y_2)-\phi(t, y_1)\Vert\le
\Vert y_2-y_1\Vert+ C_{\text{Lip}}\int_0^t\Vert\phi(s, y_2)-\phi(s, y_1)\Vert ds=R(t),
$$
and then
$
\dot R=C_{\text{Lip}} \rho\le C_{\text{Lip}} R. 
$
Gronwall's inequality implies
$$
\Vert\phi(t, y_2)-\phi(t, y_1)\Vert\le \underbrace{R\le R(0) e^{C_{\text{Lip}} t}}_{
\text{follows from Gronwall}}=\Vert y_2-y_1\Vert e^{C_{\text{Lip}} t},
$$
proving that the flow is Lipschitz-continuous with an estimate of the Lipschitz constant of the flow by the Lipschitz constant of the flux $f$ and time. Checking a linear scalar ODE proves that this estimate is essentially optimal. There are variants of this argument in the non-autonomous case.
A: $\newcommand{\norm}[1]{\lVert#1\rVert}$
$\newcommand{\lin}{\mathsf{L}}$
$\newcommand{\et}[1]{\mathsf{T}_{#1}}$
$\newcommand{\ft}{\mathsf{T}}$
$\newcommand{\bxr}[2]{\mathbb{U}(#1,#2)}$
$\newcommand{\Ck}[1]{\mathsf{C^{#1}}}$ 
By compacity, we may take $\delta>0$ such that the exponential is defined on the compact subset of $\ft M$ given by $K:=\{v\in \ft M| \,\lVert v\rVert\leq\delta\}$. The map defined on the domain of the exponential by $v_p\mapsto \norm{(\exp_p)_{\ast v_p}}_{\lin(\et{p}M,\et{\exp_p v_p}M)}$ (i.e. the operator norm of the tangent map to $\exp_p$ at $v_p$) is continuous, hence its $\sup$ on $K$, say $C$, is finite. Then, for each $p\in M$ and for each $v_p,w_p$ in the open ball $\bxr{0_p}{\delta}\subset\et{p}M$, we have $$d(\exp_p v_p,\exp_p w_p)\leq C\norm{v_p - w_p}_p.$$
To see this, take any sectionally $\Ck{1}$ curve $\gamma$ on $\bxr{0_p}{\delta}\subset\et{p}M$ joining $v_p$ to $w_p$. Then $\Gamma:=\exp_p\circ\gamma$ is a sectionally $\Ck{1}$ curve on $M$ joining $\exp_p v_p$ to $\exp_p w_p$, whose length $\ell(\Gamma)$ is $\leq C\ell(\gamma)$. Thus, $d(\exp_p v_p,\exp_p w_p)\leq\ell(\Gamma)\leq C\ell(\gamma)$. Taking the infimum over the set of sectionally $\Ck{1}$ curves on $\bxr{0_p}{\delta}$ joining $v_p$ to $w_p$ yields the asserted inequality.
