Jacobi Equation & Geodesic Deviation
Tidal Forces · Sectional Curvature · Geodesic Convergence/Divergence
Manifold
Sphere K>0
Flat K=0
Saddle K<0
Tidal Field
Curvature K
1.00
Initial separation ξ₀
0.15
Number of geodesics
7
D²ξ/ds² = −R(T,ξ)T (Jacobi equation)
Sphere K>0: ξ(s) = ξ₀ cos(√K · s)
Saddle K<0: ξ(s) = ξ₀ cosh(√|K| · s)
Focusing theorem: Ric > 0 ⟹ geodesics converge