Curvature and uniqueness of equilibrium

Let E(r) be the equilibrium manifold associated to a smooth pure exchange economy with fixed total resources r. Balasko (1980) has shown that if the equilibrium price is unique for every economy, then the price is constant, hence the curvature of E(r) is zero. By endowing E(r) with the metric induced from its ambient space, we show that, in the case of two commodities and an arbitrary number of agents, if the curvature of E(r) is zero then there is a unique equilibrium for every economy. Hence the zero curvature condition is sufficient to guarantee the uniqueness of equilibrium