I know that on a small neighbourhood about a point $p$, the exponential map is a diffeomorphism. But does the geodesic still depend continuously on the initial velocities after it has travelled a long distance from the origin?
To be precise, let $(M,g)$ be a complete Riemannian manifold, $p\in M$. For each unit length $v\in T_pM$, there is a half maximal geodesic $$\gamma_{p,v}:[0,\infty)\ni t\mapsto \gamma_{p,v}(t)\in M.$$
Now, given a number $T>0$ that can be arbitrarily large, and a sequence $\{v_n\}$ in the unit sphere of $T_pM$ such that $v_n\to v$, then do we still have that $d(\gamma_{p,v_n}(T),\gamma_{p,v}(T))\to 0$?
Of course we can write $\gamma_{p,v}(T)=\exp_p(Tv)$. However, it seems to me that $\exp_p$ is a diffeomorphism only on a small nbhd of the origin in $T_pM$, to which $Tv$ doesn't belong when $T$ is chosen large.
Is there any other way to show this? Thanks!
PS: I think we can use a series of small nbhds to cover $\gamma_{0,v}([0,T])$, argue on each, put them together and conclude.
PPS: No worries! I have already found a theorem saying that the exponential map is smooth wherever it is defined. So we don't need the diffeomorphism here!