In information Geometry, there is a definition of dual connection, which is: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfied $$X(g(Y,Z))=g(\nabla_XY,Z)+g(X,\nabla^*_XZ)$$
It seems that this definition depends on the metric $g$. My question is could we find another way to define the dual connections above, without the metric involved?