Definition . [05CN]

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Definition.

We refer to a pair of real positive definite matrix functions $V^{ij},W^{pq}$ as a solution to the split Monge-Ampère equation in an open subset $R\subset\mathbb{R}^{n}\times\mathbb{R}^{l}$ if $\det V^{ij}=\det W^{pq}$ and they are locally given by a smooth potential function $K$ :

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V^{ij}=\frac{\partial^{2}K}{\partial s_{i}\partial s_{j}},\quad W^{pq}=-\frac{\partial^{2}K}{\partial t_{p}\partial t_{q}},\quad 1\leq i,j\leq n,\quad k+1\leq p,q\leq n+l.