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 \ (g₈₉ = ᵢ ⱼ ₈₉ + H₈₉ (V, ) \) yields explicit Levi-Civita connection \ (ᵏ₈₉\), Riemann tensor \ (Rⁱ\, ₉₊₋\) with fourth-order V-derivative cancellation, Ricci tensor \ (R₈₉ (₈, ₉ ᵢ - ᵢ² ᵢ) ₈₉\), and scalar curvature \ (Kₛ ₈<₉ ᵢ ⱼ H₈₉ˢ + O (²) \). Positive/negative/zero signatures classify stability geometrically. The Lyapunov–Morse functional \ (L\) satisfies \ (d²dt²L|ₕ₍₀₄ -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 \ (0\).