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 $g$.



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> 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).
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Motivations for the study of dual connections
There may be many motivations to study dual (or maybe rather: adjoint) connections. Let me give one. The metric $g$ identifies a vector bundle $V$ and its dual $V^\*$. Any connection $\nabla$ on $V$ induces one on $V^\*$, and the pullback of this by $g$ is the dual connection. They agree if and only if $\nabla$ and $g$ are compatible.


Here is one application. If $\nabla$ is flat, this can be used to construct a characteristic class of $(V,\nabla)$, see [Bismut-Lott](http://www.ams.org/journals/jams/1995-08-02/S0894-0347-1995-1303026-5/home.html). The main idea is that $\operatorname{tr}((\nabla^\*-\nabla)^{2k+1})\in\Omega^{2k+1}(M)$ is closed, and its cohomology class is independent of $g$. These classes are global obstructions against a parallel metric with respect to $\nabla$. Note that only if $\nabla$ is flat, the local obstructions against such a metric vanishes.


