Intrinsic Geometry and the Stability of Minimum Differentiation

We develop a framework for horizontal differentiation in which firms compete on a product manifold representing the feasible combinations of characteristics. This approach generalizes both the Hotelling line and Salop circle to any Riemannian space, allowing for a unified analysis of product space. We show that equilibrium existence and stability are governed by intrinsic geometric properties, specifically curvature, symmetry and dimension. We prove that negative curvature and high intrinsic dimension act as stabilizers of minimum differentiation equilibria, moving the analysis beyond the symmetry-induced instabilities found in simpler, fixed domains. By characterizing curvature as a measure of consumer heterogeneity, we demonstrate that intrinsic geometry is a fundamental determinant of competitive outcomes.