The author investigates the space \({\mathcal D}^ 1\) of nonsingular hermitian \((n,n)\)-matrices with trace 1, equipped with the so-called Bures metric \(g^ B\) for which the natural projection \(\pi:w\to ww^*\) of the unit sphere \(S\) in \({\mathcal M}_{n,n}(\mathbb{C})\) is a Riemannian submersion. It is shown that \({\mathcal D}^ 1\) is locally isometric to the product of a sphere and the homogeneous space \(U(n)/T^ n\) with an invariant metric depending on the points of the sphere. Also the paper contains explicit formulas for the Levi-Civita connection of \(g^ B\) and its curvature.