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Information theoretic geometry near critical points in classical and quantum systems is well understood for exactly solvable systems.
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See, e.g chapter 2 of ref. [ 19 ] , and S. Kar and S. Sengupta, Pramana 69
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For a two dimensional manifold with coordinates ( x 1 , x 2 ) (x^{1},x^{2}) , R = 2 g [ ∂ ∂ x 2 ( g g 11 Γ 11 2 ) − ∂ ∂ x 1 ( g g 11 Γ 12 2 ) ] R=\frac{2}{\sqrt{g}}\left[\frac{\partial}{\partial x^{2}}\left(\frac{\sqrt{g}}{g_{11}}\Gamma^{2}_{11}\right)-\frac{\partial}{\partial x^{1}}\left(\frac{\sqrt{g}}{g_{11}}\Gamma^{2}_{12}\right)\right] , where g g is the determinant of the metric, and the Christoffel symbols are defined by Γ ν ρ μ = 1 2 g μ ζ ( ∂ g ζ ν ∂ x ρ + ∂ g ζ ρ ∂ x ν − ∂ g ν ρ ∂ x ζ ) \Gamma^{\mu}_{\nu\rho}=\frac{1}{2}g^{\mu\zeta}\left(\frac{\partial g_{\zeta\nu}}{\partial x^{\rho}}+\frac{\partial g_{\zeta\rho}}{\partial x^{\nu}}-\frac{\partial g_{\nu\rho}}{\partial x^{\zeta}}\right) . For higher dimensional manifolds, the formula is standard but more complicated. See, e.g [ 19 ]
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