Not an expert in the field. But I liked Centsov's theorem (when a metric comes from IG in the discrete case). I have not found a similar theorem for the general case. Amari's book is hard to follow. There is a serious lack of pedagogical intro to the subject, something like a starter : The main ideas + achievements of the theory + how to use it. There is something in the theory very deep but Amari just scratches the surface.

Can you imagine if GR metrics come from IG?