A Riemannian manifold $(Y,g)$ is called Monge-Ampère if it possesses an integral affine structure and the metric is, locally in affine coordinates, given by a potential $\Psi$, that is, $g_{ij}=\frac{\partial^{2}\Psi}{\partial y_{i}\partial y_{j}}$, that satisfies the real Monge-Ampère equation $\det g_{ij}=1$.