An associated bundle approach to the Bures--Wasserstein geometry of fixed rank covariance matrices

The Bures–Wasserstein geometry of covariance matrices provides a canonical distance on the statistical manifold of centred Gaussian measures and lies at the intersection of information geometry, quantum information, and optimal transport. The space of covariance matrices admits a natural stratified structure whose strata consist of fixed-rank covariance matrices. In this paper we focus on the rank-$k$ stratum ${\text{Sym}}^{+}(n,k)$ and revisit its geometry through the diffeomorphic associated-bundle model ${\text{Sym}}^{+}(n,k)\cong \text{St}(n,k){\times }_{O(k)}{\text{Sym}}^{+}(k)$. Working in this bundle picture, we (i) derive a system of differential equations for Bures–Wasserstein geodesics, (ii) prove that the fibers are totally geodesic and (iii) establish a one-to-one correspondence between Grassmannian logarithms and Bures–Wasserstein logarithms on ${\text{Sym}}^{+}(n,k)$, and hence between minimizing geodesics in the two spaces. This alternative viewpoint clarifies the role of the underlying base $\text{Gr}(k,n)$ in the Bures–Wasserstein geometry of low-rank covariance matrices and sets the stage for further investigations into structured covariance models.