Domain of the exponential map
= Domain of the exponential map
The domain of $\exp_p$ is the star-shaped open set
$$
\mathcal D_p=\{v\in T_pM:\gamma_v\text{ exists on }[0,1]\}.
$$
Smooth dependence for ordinary differential equations makes $(p,v)\mapsto\exp_p(v)$ smooth on its open domain in $TM$.