Geodesic Deviation & Jacobi Fields

D²J/ds² + R(J,γ')γ' = 0 — how nearby geodesics diverge on curved surfaces

Manifold

D²J/ds² = −K·J
J(s) = J(0)·cos(√K·s) (K>0)
J(s) = J(0)·cosh(√|K|·s) (K<0)
J(s) = J(0)·s (K=0)

Jacobi fields measure how infinitesimally neighboring geodesics spread apart. On a sphere, they focus at antipodal points. On hyperbolic space, they diverge exponentially. This is the geometric origin of tidal forces and gravitational lensing.