Bures Geodesics and Restricted Barycenters for Kronecker Positive Definite Matrices

We study the extrinsic Bures–Wasserstein geometry of the determinant-normalized Kronecker model ${\text{mcK}}_n=\{V\text{ot}U:U,V\in {\text{Sp}}^n,\det U=1\}\subset {\text{Sp}}^{n^2}$, asking when the ambient Bures geodesic between two Kronecker positive definite matrices can remain in this lower-dimensional model. Local membership near an endpoint is shown to be equivalent to membership of the whole segment, and this happens exactly in the one-factor cases: either $U_1=U_0$ or $V_1$ is a positive scalar multiple of $V_0$. Consequently, any endpoint pair not confined to these one-factor alternatives leaves the model immediately. The criterion is expressed by a partial-trace residual. In fixed commuting charts it becomes an equivalent rank-one square-root profile and yields computable departure diagnostics. We also obtain exact formulas for two restricted barycenter problems: fixed commuting-coordinate slices, solved by Perron singular vectors, and one-factor subfamilies, reduced to standard Bures–Wasserstein barycenters on ${\text{Sp}}^n$.