Recently, I have been reading through Lee's book on Riemannian Manifolds, and am currently trying to understand Jacobi fields. One of the first important theorems Lee proves in this regard is to show that any variation field through geodesics is a solution to the Jacobi equation $J'' + R(J,\gamma')\gamma' = 0$.
He then leaves it as an excercise to show that the coverse is also true. That is, given a solution to the jacobi equation, $J$, there is some variation through geodesics for which it is a variation field. More precisely, there is some variation $\gamma_s(t) = \Gamma(s,t)$ which is a geodesic for each value of s and satisfies $\frac{\partial}{\partial s}\bigr|_{s=0}\Gamma(s,t) = J(t)$.
In light of the existence and uniqueness of solutions of the Jacobi equation given the initial conditions $J(0)$ and $J'(0)$, I believe I just have to find some variation which will satisfy these initial conditions for arbitrary choices of $J(0)$ and $J'(0)$.
Lee suggests as a hint the general form $\exp_{\sigma(s)}(tW(s))$ for some curve $\sigma$ in $M$ and some $W(s) \in T_{\sigma(s)}M$, and I am aware that $\Gamma(s,t) = \exp_{\gamma(0)}(t(\gamma'(0) + sJ'(0)))$ suffices for the special case $J(0) = 0$
Having played around a little, it seems that the condition for $J(0)$ aught to come from $\sigma$, although I am unsure of what type of curve should suffice for $\sigma$, as I don't quite know how I should interpret the derivative $\frac{\partial}{\partial s}\bigr|_{s=0}\exp_{\sigma(s)}(v)$.
What curve $\sigma$ would give the desired initial conditions, and might there be any hints how to better understand how one takes a derivative along the base of the exponential function as above?