A Monge-Ampère manifold is a triple $(Y,g,\nabla)$, where $(Y,g)$ is a smooth Riemannian manifold with the metric $g$, and $\nabla$ is a flat connection on $T_{Y}$ such that:

a) $\nabla$ defines an affine structure on $Y$.

b) Locally in affine coordinates $(x_{1},...,x_{n})$ the matrix $((g_{ij}))$ of $g$ is given by $((g_{ij}))=((\partial^{2}K/\partial x_{i}\partial x_{j}))$ for some smooth real-valued function $K$.

c) The Monge-Ampère equation $det((\partial^{2}K/\partial x_{i}\partial x_{j}))=const$ is satisfied.