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,

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!