I am intrigued by the notion of dual connections: two affine connections $\nabla$ and $\nabla^*$ are called dual if they satisfy $$X(g(Y,Z))=g(\nabla_XY,Z)+g(Y,\nabla^*_XZ)$$ for a given (pseudo)-riemannian metric $g$.

What is the motivation and the deep results behind this notion?

What are the main fields of application: information geometry, riemannian foliations, webs ...?

I am interested in any nice reference or survey paper (especially any information geometry paper written by a 'true' mathematician).