  The Jordan product on the self-adjoint part of a finite-dimensional
$C^{*}$-algebra $\mathscr{A}$ is shown to give rise to Riemannian metric
tensors on suitable manifolds of states on $\mathscr{A}$, and the covariant
derivative, the geodesics, the Riemann tensor, and the sectional curvature of
all these metric tensors are explicitly computed. In particular, it is proved
that the Fisher--Rao metric tensor is recovered in the Abelian case, that the
Fubini--Study metric tensor is recovered when we consider pure states on the
algebra $\mathcal{B}(\mathcal{H})$ of linear operators on a finite-dimensional
Hilbert space $\mathcal{H}$, and that the Bures--Helstrom metric tensors is
recovered when we consider faithful states on $\mathcal{B}(\mathcal{H})$.
Moreover, an alternative derivation of these Riemannian metric tensors in terms
of the GNS construction associated to a state is presented. In the case of pure
and faithful states on $\mathcal{B}(\mathcal{H})$, this alternative geometrical
description clarifies the analogy between the Fubini--Study and the
Bures--Helstrom metric tensor.
