I want to show that every Jacobi field is a variation of geodesics, i.e. let $Y: I \rightarrow TM$ be a Jacobi field along a geodesic $\gamma$, then I want to show that $Y$ can be written as a variation of geodesics.
1.) It is clear that we only need to show that the initial conditions of the variation agree with the field, cause a variation always satisfies the Jacobi equation. The uniqueness of 2nd order ODEs implies then the theorem.
2.) My idea was that such a variation might look like $\Gamma(s,t):=\exp_{\alpha(s)}(t V(s)),$ where I want to specify $\alpha$ and $V$ later on.
Now, $\partial_s|_{s=0}\Gamma(s,0)= D \exp_{\alpha(0)}(0)(\alpha'(0))= \alpha'(0).$
This might be interesting, as it suggests the choice $\alpha'(0):=Y(0).$
Now, we would also like to have $\nabla_{\gamma'(t)}|_{t=0} \partial_s|_{s=0}\Gamma(s,t) := \nabla_{\gamma'(t)}|_{t=0}Y = Y'(0).$
But I don't see how we can actually construct all this and what conditions this gives us on $\alpha$ and $V$?. Does anybody know how this theorem can be shown?
If anything is unclear, please let me know.
In case you are interested: I actually found a proof, but I have troubles understanding one step Covariant and partial derivative commute?