I stumbled over the question: Given a manifold $M$. Can there exist a vector field that is a Jacobi field along every geodesic?
Now, the answer is apparently: Yes, because the $0$ vector field does it. But I started thinking whether there could be other non-trivial vector fields along every geodesic.
My intuition tells me that if the dimension of the manifold is larger or equal to 2, then the statement should be false(but don't know of a proof), but for 1d manifolds at least, this might be sometimes possible. (For example, if the manifold is defined by a geodesic).
Yes. There could be for some manifold. For example if you have a Killing vector fields $X$ on $M$, then the diffeomorphism group generated by $X$ are isometries. Thus it sends geodesics to geodesics. It is known that such a vector fields are Jacobi fields (along any geodesics).
For example, any constant vector fields on $\mathbb R^n$ and flat torus are such an examples. Also vector fields generated by roatations of $\mathbb S^2$.