Fisher-Rao Metric on the Gaussian Manifold
The family of Gaussian distributions N(μ,σ²) forms a 2D Riemannian manifold. The Fisher information metric is: ds² = dμ²/σ² + 2dσ²/σ². This is the Poincaré upper half-plane metric — the space has constant negative curvature K=−½, making it hyperbolic! The geodesic distance (KL-based) between N(μ₁,σ₁²) and N(μ₂,σ₂²) differs from Euclidean distance. The manifold map shows iso-μ and iso-σ curves, geodesics (semi-circles in the half-plane), and the KL divergence heatmap from Gaussian 1.