@BenMcKay Yes assume I know that $DG$ has full rank at all points, so we cna apply the implicit function theorem, so we can express the level set locally as a graph.

Thank you! I'll search for them. Do you know any other European countries where mathematicians might be doing the same thing?

By the way, I'm not working in mathematics itself. I'm working in medical imaging, and just need the formula rather than the derivation. I'd be interested in the derivation too, but just afraid that it'd take a long time for me off the track, although it'd certainly be a great exercise!

Thank you Stafan Waldmann. I'd definitely love to do this as an exercise. But what I need it is only a concrete formula to apply in a medical imaging problem, so the detailed derivation could be skipped for the moment.

I'm not quite sure yet! Because here both $t$ and $s$ are varying. I mean if you keep $t$ fixed and vary $s$, i.e. for ONE Jacobi field, I totally agree. But if you vary both $t,s$, I've doubt. Here the Jacobi fields are varying with time. It's like we're considering scalar functions $f_t(s)$, expanding w.r.t. $t$, and then set $s=2t$, which I'm not sure will give Taylor expansion of $f_t around $t$$. Here we're expanding $f_t(2t)$ w.r.t. $t$. Could you please either prove your formula OR give a reference? I've known Taylor expansions of norms of Jacobi from Do Carlo, but not Jacobi itself.