Curvature of the manifold of fixed-rank positive-semidefinite matrices endowed with the Bures–Wasserstein metric

Massart, Hendrickx, and Absil compute sectional curvature on the fixed-rank PSD quotient, and riva2026random reuses their formulas to show that curvature on the base manifold blows up when the two smallest eigenvalues of P collapse together, tying holonomy strength to the spectral gap.