Suppose we have two variation vector fields through geodesics, $J_1$ and $J_2$, along a geodesic $\gamma_0$. Then $J_1$ and $J_2$ are Jacobi fields, but I do not think that is imporant for what I am looking for. Specifically, if $h_1(s,t)$ and $h_2(s,t)$ are the variations, with $s\mapsto h_i(s,t)$ the transverse curves and $t\mapsto h_i(s,t)$ the longitudinal curves, then \begin{align} J_i = \left.\frac{d}{ds}\right|_{s=0}h_i(s,t), \end{align} $\gamma_0(t) = h_i(0,t)$, and $h_i(\epsilon,t)$ define nearby geodesics, infinitesimally so in the limit $\epsilon \to 0$. In physics we often imagine $J_i$ to be "pointing to" some infinitesimally nearby geodesic $\gamma_i$, and I wonder if it is possible to construct a third variation $h_3$, but this time "along $\gamma_2$," which is such that $J_3$ "points to" $\gamma_1$?
Formally, I would like to imagine a construction that defines $h_3(s,t)$ such that $h_3(0,t) = h_2(\epsilon,t)$ and $h_3(\epsilon,t) = h_1(\epsilon,t)$, and then take the limit $\epsilon \to 0$. However, does that even make sense? We could try to claim that we get the components \begin{align} J^\mu_3 = \lim_{\epsilon \to 0}\frac{h_1^\mu(\epsilon,t) - h^\mu_2(\epsilon,t)}{\epsilon}, \end{align} in some local chart, but there does not seem to be a concrete tangent space where $J_3$ lives. Is this a problem (I believe it is)? Can we remedy it somehow?
To be concrete: I am interested in finding $J_3$ given $J_1$ and $J_2$, but I am not sure that we can even speak of $J_3$ as existing; I am having trouble defining what we should actually require of $J_3$ in a meaningful sense.