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.
在三维空间中，所有瑟斯顿几何的“主理论”都需要虚通量。然而，这些几何可以从物理三维理论与各种额外的标量场，这可以被解释为模在各种紧致化的高维“主理论”。三个瑟斯顿几何的形式为N_2 x S^1，其中N_2表示常曲率的二维黎曼空间。这使我们能够通过T-对偶将这些空间扭曲成N_2上的U（1）丛的其他Thurston几何。这样，在高维M-理论框架中，霍普夫T-对偶关系除了一个几何之外的所有几何。例外的是“溶胶几何”，这是从D3-膜在背景B-场的去耦极限降维的结果。
