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?



Local Lipschitz constant of exponential map for Hadamard manifolds
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
$$
 |d\exp\_x(v)| \le \frac{\sinh \kappa|v|}{\kappa|v|}.
$$


