Let $(M, g)$ be a complete, smooth Riemannian manifold. $\forall p \in M, \ \exists U \subset M, \ p \in U ,$ s.t $exp^{-1}_{p}|_{U}$ is a diffeomorphism, and so $exp^{-1}_{p} $ is smooth on $U$.
My question is whether this property holds globally on a complete riemannian manifold. Is the inverse exponential map $exp^{-1}_{p} : M \rightarrow T_pM$ a smooth map for all $p \in M$ ?
If not, under what additional assumptions would this be true ?