Curved surfaces · parallel transport · geodesic deviation · Gaussian curvature
Sectional curvature K(σ) = R(u,v,u,v)/|u∧v|² for a 2-plane σ spanned by vectors u,v. For a sphere: K = 1/R² > 0 — geodesics converge (great circles meet at poles). For a saddle: K < 0 — geodesics diverge exponentially (Hadamard-Cartan theorem). Gauss-Bonnet: ∫∫K dA + ∮κ_g ds = 2πχ(M). Geodesic equation: ∇_γ̇ γ̇ = 0. Colors show Gaussian curvature K at each point.