Local Lipschitz constant of exponential map for Hadamard manifolds Suppose that $(M,X)$ is a simply connected complete Riemannian manifold with pinched sectional curvature between $[a,0]$.  Let $r>0$ and fix any point $p\in M$.  Is there a bound on the local Lipschitz constant of the Riemannian exponential map $\exp_p$ restricted to the Euclidean ball at the origin of radius $r$, written in terms of these sectional curvature bounds?
 A: Here is an argument that gives a sharp estimate.
Given $x \in M$, $v \in T_xM$, and $(e_1, \dots, e_n)$ an orthonormal basis of $T_xM$,
$$
  (d\exp_x(v))e_k = J_k(1),
$$
where $J_k$ is the Jacobi field along the constant speed geodesic
$$(t) = \exp_x(tv),\ 0 \le t \le 1,
$$
that satisfies $J_k(0) = 0$ and $\nabla_{c'}J_k(0) = e_k$. Therefore, it satisfies the Jacobi equation
$$
  \nabla^2{c'c'}J_k = R(c',J_k)c'.
$$
Parallel transport $(e_1, \cdots, e_n)$ along the curve $c$ and let $J(t)$ be the matrix such that along $c$,
$$
  J_k = e_jJ^j_k
$$
From the Jacobi equation, it follows that
[
J'' + KJ = 0,
]
where $K$ is the symmetric matrix given by
$$
  K_k^j = e_j\cdot R(e_k,c')c'.
$$
If the sectional curvature is bounded from below by $-\kappa^2$, then
$$
  K \ge -\kappa^2|v|^2 I,
$$
because $|c'| = |v|$.
Let $A = J'J^{-1}$, which satisfies
\begin{align*}
  A' + A^2 + K &= 0,\ 0 \le t \le 1\\
  A(t) &= t^{-1}I + O(t)\\
  A^T &= A.
\end{align*}
As an aside, $A$ is the second fundamental form at $\exp_xv$ of the geodesic sphere of radius $|v|$ centered at $x$ written with respect to the frame $(e_1, \dots, e_n)$.
If
$$
  a(t) = \frac{\kappa|v|\cosh \kappa |v|t}{\sinh \kappa |v|t},
$$
then
\begin{align*}
  a' + a^2 - \kappa^2|v|^2 = 0.
\end{align*}
If $B = A-aI$,
then a straightforward calculation shows that
\begin{align*}
  B' &\le -2aB.
\end{align*}
It is easy to check that
\begin{align*}
  \lim_{t\rightarrow 0} B(t) &= 0.
\end{align*}
It follows that $B \le 0$, i.e., $A \le aI$., Therefore,
\begin{align*}
  |J|' &= \frac{J\cdot J'}{|J|} = \frac{JAJ}{|J|}
       \le a|J|.
\end{align*}
This implies that
$$
  |J|(t)| \le \frac{\sinh \kappa|v|t}{\kappa |v|}.
$$
This gives the sharp estimate that if the sectional curvature is bounded from below by $-\kappa^2$, where $\kappa > 0$, then
$$
  |d\exp_x(v)| \le \frac{\sinh \kappa|v|}{\kappa|v|}.
$$
