a Jacobi field
/ yah-KOH-bee /
Fire two arrows in almost the same direction from the same bow; at first they stay close, but on a curved range the gap between them grows or shrinks in a way that flat ground would never show. A Jacobi field is the precise record of that spreading. It is the infinitesimal separation between a geodesic and a nearby geodesic that started just slightly differently — the velocity at which a family of geodesics fans apart — and it is the single most important tool for turning a local curvature bound into a global statement about distance.
Make it precise with a variation. Take a smooth family of geodesics gamma_s(t), one for each small s, all starting near a base geodesic gamma_0(t) = gamma(t). The variation field J(t) = (d/ds)|_{s=0} gamma_s(t) measures, at each time t, how fast neighbours pull away. Such J always satisfies the Jacobi equation, a linear second-order ODE along gamma: J'' + R(J, gamma') gamma' = 0, where '' is the covariant derivative along gamma and R is the curvature tensor. Read it as Newton's law for the gap: curvature is the force. Where sectional curvature K is positive the curvature term pulls neighbouring geodesics back together (J oscillates, like sin); where K is negative it pushes them apart (J grows, like sinh); where K = 0 they drift linearly. The solution space is 2n-dimensional, fixed by J(0) and J'(0).
Jacobi fields are the engine of comparison geometry. A conjugate point is exactly a parameter value t* > 0 where a nontrivial Jacobi field with J(0) = 0 returns to zero, J(t*) = 0 — two infinitesimally separated geodesics from p reconverge — and past the first such point the geodesic stops minimizing. The Rauch comparison theorem bounds the size of |J| by comparing the actual curvature to a constant-curvature model, and that single estimate underlies Bishop-Gromov volume comparison, Toponogov's theorem, and the sphere theorem. Caveat: the curvature sign convention in J'' + R(J, gamma') gamma' = 0 must match your R; with the opposite sign convention the equation reads with a minus, so always check the book's convention before trusting a formula.
On the unit sphere S^2 (constant K = 1), two geodesics leaving the north pole are great circles; the Jacobi field measuring their separation has magnitude |J(t)| = (sin t) times its initial spread rate, because the Jacobi equation reduces to J'' + J = 0. It grows, peaks at the equator (t = pi/2), then shrinks back to zero at the south pole t = pi — that vanishing is exactly why the south pole is conjugate to the north pole, and why geodesics stop minimizing there. On flat R^2 the same field is |J(t)| ~ t (the equation is J'' = 0): neighbours drift apart linearly and never reconverge.
Sphere (K=1): |J| = sin t, vanishes at the conjugate south pole. Plane (K=0): |J| grows linearly, never returns.
A Jacobi field is not any old vector field along a geodesic — it must solve the Jacobi equation. The field tangent to gamma itself (J = t gamma') and the constant field gamma' are always trivial Jacobi fields; the interesting ones are the normal components.