The Fisher information metric makes the space of probability distributions a Riemannian manifold. Geodesics correspond to paths of minimum information loss. For Gaussians, this is the Poincaré upper half-plane (constant negative curvature).
Left: Gaussian distributions in x-space. Right: Statistical manifold of Gaussians — points are (μ,σ) and the Fisher metric ds²=dμ²/σ²+2dσ²/σ² makes this the Poincaré half-plane (K=−½). Geodesics are semicircles/vertical lines. KL divergence shown.