Riemannian Manifolds of Asymmetric Equilibria: Complete Geometric Structure and Curvature Signatures

We develop the complete Riemannian geometry of Victoria–Nash asymmetric equilibrium manifolds (VNAE) for n-player games. The metric \(gᵢⱼ = ι_i ι_j δᵢⱼ + ε Hᵢⱼ(V,ι)\) yields explicit Levi-Civita connection \(Γ^kᵢⱼ\), Riemann tensor \(R^i\,jkl\) with fourth-order V-derivative cancellation, Ricci tensor \(Rᵢⱼ ≈ κ(ιi,j ι_i - κ ∂_i^2 ι_i) δᵢⱼ\), and scalar curvature \(K_s ≈ ∑i<j ι_i ι_j Hᵢⱼ^s + O(ε^2)\). Positive/negative/zero signatures classify stability geometrically. The Lyapunov–Morse functional \(L\) satisfies \(d^2/dt^2L|VNAE ≈ -2 Ric(ṡ^⊥,ṡ^⊥)\) along gradient flows, establishing Ricci curvature as the normal contraction rate. Classical Nash, von Neumann’s minimax theorem, and Lyapunov stability emerge as degenerate flat limits as \(ε→0\).