Shortest paths on curved surfaces — geodesic equation, Gaussian curvature, and Jacobi field deviation
Geodesic Equation
The geodesic equation — shortest paths on a curved manifold:
ẍᵏ + Γᵏᵢⱼ ẋⁱ ẋʲ = 0
where Γᵏᵢⱼ are the Christoffel symbols, encoding how the metric varies:
Γᵏᵢⱼ = ½ gᵏˡ(∂ᵢgⱼˡ + ∂ⱼgᵢˡ − ∂ˡgᵢⱼ)
Sphere: great circles are geodesics (θ̈ − sin θ cos θ φ̇² = 0)
Torus: latitudinal circles at the outer equator are geodesics; inner circles are unstable. Meridional circles are always geodesics.
Gaussian Curvature & Jacobi Fields
Gaussian curvature K = κ₁κ₂ (product of principal curvatures).
Sphere: K = 1/R² everywhere (positive → geodesics converge)
Torus outer: K > 0 (sphere-like, converging)
Torus inner: K < 0 (saddle-like, diverging)
Jacobi fields J(t) measure geodesic deviation — how neighboring geodesics spread:
J̈ + K(γ(t))·J = 0
K > 0 → oscillating J (focusing, conjugate points)
K < 0 → exponentially growing J (defocusing, no conjugate points)