Jacobi field

Summary

A Jacobi field is a vector field along a geodesic that solves the Jacobi equation, a linear second-order equation involving the Riemann curvature tensor. Jacobi fields are exactly the variation fields of families of geodesics, so they describe how nearby geodesics spread apart or come together. They connect curvature with the behaviour of geodesics and lead to conjugate points, comparison theorems and the second variation of length.