This paper shows that simple geometric properties of the equilibrium set of a market force equilibrium prices to stop varying. The authors study the equilibrium manifold E(r) that collects price–endowment pairs for economies with fixed aggregate resources r. They prove that when E(r) is intrinsically flat — meaning its Riemann curvature tensor is zero, so it has no internal geometric bending — the equilibrium price vector is locally constant. In plain terms, a flat equilibrium surface cannot support small changes in the equilibrium price.

The work treats a smooth pure exchange economy with normalized prices and arbitrary numbers of commodities and consumers. Using the metric induced from the ambient Euclidean space, the authors prove a general result: intrinsic flatness of E(r) implies the Jacobian of the equilibrium price map is zero, so prices do not change to first order when endowments move. Paired with Balasko’s uniqueness–constancy criterion (from earlier literature), this gives a necessary and sufficient condition: E(r) is intrinsically flat if and only if the normalized equilibrium price is unique for every economy with the same aggregate resources r. This extends previous results that covered only low-dimensional cases.

The proofs use a standard local parametrization of E(r) that treats some endowment coordinates as free and others as residual. That parametrization is affine in the free endowment variables. A key linear-algebra fact — the “tangential lemma” in the paper — shows that if certain canonical endowment directions lie in the tangent space, then the differential (Jacobian) of the price map must be zero. The authors then apply the Gauss equation, which links intrinsic curvature to the second fundamental form (the way the surface bends in the ambient space). Because of the chosen parametrization, some components of the second fundamental form vanish, and intrinsic flatness forces the remaining mixed components to vanish too. This route avoids building an explicit normal frame or computing full curvature tensors.