I am reading an introductory textbook in Riemannian geometry (Lee), and met this problem:
Any Jacobi field $V$ along a geodesic segment $\gamma$ is the variation field of some variation of $\gamma$ through geodesics.
A hint is given: try to write the family $\Gamma (s,t)$ as $\exp_{\sigma(s)}tW(s)$ for some proper curve $\sigma(s)$ and vector field $W(s)$ along $\sigma$. This is reasonable since for each $s$, $\Gamma (s,t)$ gives a geodesic.
The problem is of course how to find such $\sigma$ and $W$. Clearly we should require $\dot \sigma (0) = V(0)$, and $W(0)$ be determined by the original geodesic $\gamma$. But what's next? I plan to find $\partial / \partial t \ \Gamma $ and $\partial / \partial s\ \Gamma $, and see whether I can get something from the Jacobi equation. But I do not know how to differentiate $\exp$ with respect to $\sigma(s)$!
Anyone has more hint(s)? Thank a lot!