Intrinsic Fourier Geometry of Gaussian Laws

On the manifold of nondegenerate Gaussian laws we study the intrinsic length metric induced by the supremum norm of differences of characteristic functions. The Fourier embedding gives a reversible location–covariance tangent norm. In one dimension, its radial maximization yields an explicit Finsler model and globally minimizing location–scale geodesics, including a closed integral expression for equal endpoint variances and a unique scalar equation for general endpoints. The equal-variance distance is strictly increasing and concave and has logarithmic growth with an explicit constant. In arbitrary dimension, a Gaussian block carrying both mean and covariance variation combines with a covariance-only block by an exact maximum of intrinsic distances. This gives a spectral/Krylov reduction and precise formulas for relative-covariance eigenvector classes. A reflection argument also determines the exact distance to the set of Gaussians with prescribed terminal mean and freely varying terminal covariance. We establish existence and nonuniqueness results for minimizers. The direct Fourier metric, the intrinsic metric, and the classical Thompson covariance geometry are kept distinct.