Jacobi field visualisation

Question

Are there any good pictures or applets representing how Jacobi fields depend on their initial vectors? The textbook I'm using told me that solutions exist based on ODE theory, but I'm not sure how to visualise them.

Answer 1

There's a nice picture in John M. Lee's Riemannian Manifolds: An Introduction to Curvature (https://www.maths.ed.ac.uk/~v1ranick/papers/leeriemm.pdf), Figure 10.4. Lee states that the set of Jacobi fields along a geodesic is a $2n$-dimensional linear space (Corollary 10.5), with 2 tangential dimensions and $2n-2$ normal dimensions. These correspond to independent, arbitrary choices of $J(0) \in T_p (M)$ and $D_t J(0) \in T_p M$. It seems like tangential components are "trivial", i.e. you get a reparametrisation of the same geodesic.

In all the normal directions, the the simple thing you can do is "rotate the geodesic in that direction" (see Lemmas 10.7 and 10.8, as well as the figure). I haven't found a good picture or description of the other direction, but I suspect you can essentially "move the geodesic in space" in that direction. If you're on a sphere, rotating the geodesic would give you a conjugate point at distance $\pi$, while moving the geodesic would give you a conjugate point at distance $\frac{\pi}{2}$.