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?
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> What are the main fields of application: information geometry, riemannian foliations, webs ...?
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> I am interested in any nice reference or survey paper (especially any information geometry paper written by a 'true' mathematician).