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 $TY$ 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 $(g_{ij})=(\partial^{2}F/\partial x_{i}\partial x_{j})$ for some smooth real-valued function $F$.

c)

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