Curvature, Minimality and Uniqueness of Equilibrium

For a smooth pure exchange economy with fixed aggregate resources, we study two geometric conditions on the equilibrium manifold $E(r)$ endowed with the metric induced from its Euclidean ambient space. First, for arbitrary numbers of commodities and consumers, we prove that intrinsic flatness forces equilibrium prices to be locally constant. Together with Balasko's uniqueness--constancy criterion, this yields a necessary and sufficient condition: $E(r)$ is intrinsically flat if and only if the normalized equilibrium price is unique for every economy with aggregate resources $r$. This extends the curvature--uniqueness theorem of \cite{LoiMatta2018} and completes the higher-dimensional direction pursued in \cite{LoiMattaUccheddu2023}. Second, in the two-commodity case, we show that minimality of $E(r)$ already forces local constancy of the price map. Under the uniform-distribution interpretation of \cite{LoiMatta2021}, this gives the minimal-entropy/uniqueness equivalence without the additional asymptotic assumption used there. Both arguments rely on the same local parametrization of $E(r)$ and avoid the explicit construction of a normal frame.