Statistical manifolds, geodesics, and the geometry of probability distributions
The Fisher-Rao metric g_ij = E[∂_i ln p · ∂_j ln p] gives the unique Riemannian metric on statistical manifolds (Chentsov 1975). For Gaussians N(mu,sigma²), the manifold is the Poincaré upper half-plane with hyperbolic metric ds² = (dmu² + 2dsigma²)/sigma². Geodesics are semicircles perpendicular to the mu-axis.