I apologize if this is not a research level question (already tried asking <https://math.stackexchange.com/questions/1303288/relation-between-parallel-transport-and-jacobi-field-iion> stack exchange with no response), but it's not a homework question either: an applied mathematics/medical imaging paper I was reading tried to numerically approximate parallel transport at time step $i$ using Jacobi fields, and when I ran their algorithm and checked the norms of the parallel transport, it's blowing up (literally) exponentially as a function of the time steps $i$ I used to divide $[0,1]$, which tells me it might not be correct.


Here's the the theoretical assumption/question regarding their algorithm:


FIX $t\_0>0$.


Let $t>0$ and $W(t\_0)\in T\_{c(t\_0)}M$ be the parallel transport of $W\in T\_{c(0)}M$. and $J\_{t\_0,W(t\_0)}$ be the Jacobi field along a geodesic $c$ such that $J\_{t\_0,W(t\_0)}(t\_0)=0, J\_{t\_0,W(t\_0)}'(t\_0)=W(t\_0)$. Let $P\_{t\_0,t\_0+t}$ denote the parallel transport from time $t\_0$ to $t\_0+t$ for a FIXED $t\_0\in \mathbb{R}$


By now, one can prove that:(I agree and I've the proof)


$$\left|\left|\frac{P\_{t\_0,t\_0+t}(W(t\_0))-\frac{J\_{t\_0,W(t\_0)}\ \ \ (t\_0+t)}{t}}{t}\right| \right|\to0$$ 
as $t\to 0$.


Now, let $t\_0$ VARY, and indeed assume that $t\_0=t$. My question is: is the same formula going to be true? That is, do we also have:


$$\left| \left|\frac{P\_{t,2t}(W(t))-\frac{J\_{t,W(t)}\ \ \ (2t)}{t}}{t}\right| \right|\to0$$
as $t\to 0$? THIS IS MY MAIN QUESTION. The paper seemingly used this. A definite yes or no answer is okay with me for the moment. Proofs very much appreciated.


In the above, $J\_{t,W(t)}$ is the unique Jacobi field along $c$ so that $J(t)=0, J'(t)=W(t)$.


In that paper I mentioned, they assume $t=\frac{1}{N}$and $P\_{t,2t}(W(t))=P\_{0,2t}(W)$ is approximately $\frac{J\_{t,W(t)}(2t)}{t}$, and use it to repeatedly calculate Jacobi and parallel transport. But the norms of $P\_{t,2t}(W(t))$ is showing huge errors and $P\_{t,kt}(W(t))$ is blowing up with $k$.



Using Jacobi fields to approximate parallel transport along geodesic:is the following limit true?
The answer is yes. Since the Jacobi field satisfies the second order equation $$ J(s)'' + R(s)J(s)=0,$$ where $R(s)$ is the curvature tensor (details are skipped here); then for $J\_{t,W(t)}(s)$ it holds $J(s)=(s-t)W(s) + (s-t)^3 R(s)W(s)/6$ up to higher orders of $(s-t)$ as $s-t$ goes to 0 ($J(t)=0$ is used here). So, if the curvature tensor $R$ of $M$ behaves nicely (say, is bounded) the approximation should work. 


