Let $M$ be a complete (connected) Riemannian manifold, $p \in M$ some point in $M$. Assume $exp_p$ is a diffeomorphism from $T_pM$ onto $M$.
Is it true that $exp_q$ is a diffeomorphism for all points $q \in M$?
Of course if $M$ has a transitive isometry group, than the answer is positive, but what about other cases?
Note that according to this answer this is equivalent to asking whether all geodesics of $M$ are globally length minimizing or all points in $M$ are joined by unique geodesics.