Let $y_{1},\ldots,y_{n}$ be local affine coordinates on $B$ with respect to the affine structure induced by $\omega$. Then locally there is a function $K$ on $B$ such that

$$g(\partial/\partial y_{i},\partial/\partial y_{j})=\partial^{2}K/\partial y_{i}\partial y_{j}.$$

Furthermore, $\check{y}_{i}=\partial K/\partial y_{i}$ form a system of affine coordinates with respect to the affine structure induced by $\operatorname{Im}\Omega$, and if

$$\check{K}(\check{y}_{1},\ldots,\check{y}_{n})=\sum\check{y}_{i}y_{i}-K(y_{1},\ldots,y_{n})$$

is the Legendre transform of $K$, then

$$y_{i}=\partial\check{K}/\partial\check{y}_{i}$$

and

$$\partial^{2}\check{K}/\partial y_{i}\partial y_{j}=g(\partial/\partial\check{y}_{i},\partial/\partial\check{y}_{j}).$$