How solution of Jacobi field indicate the behaviour of nearby geodesics I was learning Riemann geometry,we know that positive curvature affects the behavior of nearby geodesics converge,taking the sphere for example, we have the Jacobi field for the sphere that vanishing at $t = 0$ of the form $$J(t) = k\sin(t)E$$
for some parallel unit normal vector field $E$.
For negative curvature we have solution of the form $$J(t) = k\sinh(t)E$$
Why the solution for the Jacobi fields(sign of curvature) indicate the behavior of nearby geodesics converges/diverge?What's the intuition of the Jacobi field here?
 A: First, it is easy to see how geodesics behave on constant curvature space directly from the geometric models of the sphere and hyperbolic space. But it's hard to infer this behavior directly from the nonlinear ODE satisfies by a geodesic and how curvature affects the geodesics. This becomes even more challenging when the curvature is not constant.
A Jacobi field is an infinitesimal variation of a geodesic. It satisfies a linaer ODE, which is an infinitesimal variation of the geodesic ODE. This linear ODE turns out to be much more useful, because the curvature appears explicitly, and it turns out to be a self-adjoint second order ODE, which can be analyzed using Sturm-Liouville theory. In essence, the ODE is of the form
$$ J'' + JK = 0, $$
where $K$ is a piece of the curvature tensor. This is easily solved when $K$ is constant, which confirms what we already know about the sphere and hyperbolic space. But what's even better is that we can now use, for example, the Sturm comparison theorem to infer results when curvature varies but satisfies certain bounds. Also, $J'J^{-1}$ satisfies a matrix Riccati equation, which also can be analyzed using ODE techniques to study the geometric and topological properties of a Riemannian manifold.
