Jochen Dittmann: On the Riemannian geometry of finite dimensional mixed states Journal of Lie Theory, vol. 3 (1), p. 73-87 We consider the Riemannian geometry of the space of nonsingular density matrices $\cD^1$ equipped with the Bures metric $g^B$. This space is of certain physical relevance on the background of a generalization of the Berry phase to mixed states. We determine the covariant derivative and the curvature tensor field related to the Levi-Cevita connection of $(\cD^1,g^B)$, which allow us to calculate other curvature quantities . It turns out that $\cD^1$ is not a space of constant curvature and not even a locally symmetric space, in contrast to what the case of two-dimensional density matrices might suggest. Moreover, we give a local description of $\cD^1$ and explicit formulae for the Bures metric in terms of natural matrix operations containing $\r$ and $\D\r$ only.