If the Exponential map is a diffeomorphism at a point, can we say something about other points?

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.

As an example (without being rigorous) consider a manifold which looks like an inifinite half cylinder parallel to the positive $z-$axis in Euclidean three space with a hemisphere attached at the bottom along an equator and smoothed out. (A bit like a hyperboloid or an one sided infinite cigar). If you look at the south (bottom) pole of the hemisphere you will get your diffeomorphic exponential map (the geodesics starting from there will move from there to the cylinder and then parallel to the z-axis along the cylinder). But if you look a a point on the cylinder, you will even find a closed geodesic running around the cylinder. (I hope I made this clear enough by describing it just in words).
• Great example!${}$ – user98602 Nov 28 '15 at 15:32