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Apologies if this is not exactly a research-level questions, but I've no known reference where I can figure it out myself. I asked this on math.stackexchange.com,

https://math.stackexchange.com/questions/1309122/taylor-expansion-of-riemannian-exponential-map-and-jacobi-fields

with no replies.

1) Let $(M,g)$ be a Riemannian manifold, and at $p\in M,$ let $exp_p:U\subset T_pM\to M$ denote the Riemannian exponential map defined on $U\subset T_pM.$ I'd like to know how I could expand $t\mapsto exp_p(tv)$, possibly with the first term $exp_p(0)=p$. Now, I know that addition would NOT make sense unless you assume that $M\subset \mathbb{R}^n$, so assume whatever you need to assume topologically or as differential manifolds, however, $M$ is NOT isometrically embedded in $\mathbb{R}^n$.

2) Secondly, with OR without any such assumption above, how can I expand the Jacobi field $J(t)$along a geodesic $c(t)\subset M$ so that $J(0)=0$ around $t=0$ ?. We know $J(t)=\frac{\partial}{\partial\epsilon}|_{\epsilon=0}exp_p(v+\epsilon w)=(Dexp_p)_{tv}(tw)$. Can I use the Taylor expansion of the exponential map to obtain that of its derivative? I'm hoping for an expression involving first term zero, second term involving $J'(0)=W,$, third term zero (as the second derivative $J''(0)=0$), the fourth term involving the curvature term $K(v,w)$, possible some parallel translates $P_{0,t}$ of certain vectors, and then higher order terms. I've seen a concrete such expansion for $||J(t)||$in Do Carmo's book, chapter on Jacobi fields. But I've never seen one for $J(t)$ itself.

A detailed answer OR at least a suitable reference (or a book which gives this as an exercise with ample hints) would be highly appreciated!

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closed as off-topic by Deane Yang, Alex Degtyarev, Stefan Kohl, Stefan Waldmann, Daniel Moskovich Jun 22 '15 at 20:27

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    $\begingroup$ Yes, please migrate to math.exchange.com. Two hints: use Jacobi equation and describe differential of exponential in terms of Jacobi fields. Solid exercise for a differential geometry student. Might take you a while but you'll learn more than just reading about it. $\endgroup$ – Deane Yang Jun 22 '15 at 12:44
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    $\begingroup$ Oops. I didn't see that you already posted this on math.stackexchange.com. But I stand by my other advice. If you dig around MathOverflow, I believe you'll also find other answers to this question. $\endgroup$ – Deane Yang Jun 22 '15 at 13:41
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    $\begingroup$ You can find this in an appendix to my lecture notes on wave equations. But as Deane said: good exercise to do it yourself :) $\endgroup$ – Stefan Waldmann Jun 22 '15 at 16:26
  • $\begingroup$ 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. $\endgroup$ – Let's talk math Jun 22 '15 at 17:48
  • $\begingroup$ 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! $\endgroup$ – Let's talk math Jun 23 '15 at 9:58