Two Koszul connections D and D* are conjugate with respect to (pseudo)-riemannian metric g if:
X(g(Y,Z))=g(D_XY,Z)+g(Y,D*_XZ).
What is the motivation and the deep results behind this notion?
I encountered this notion in geometry of information papers, but I have not studied them in detail.
1) The connection D* in the question should be denoted by D*_g !
2)A connection D in a fiber bundle \pi canonically induces connections in all tensor powers of this bundle, in particular, a connection D* in the dual bundle. Informally speaking, if one moves according the connection a fiber of this bundle, then he moves automatically any its tensor power.
3) A nodegenerate second order covariant tensor g, not necessarily a pseudo-metric, identifies \pi and \pi^*. For instance, Z --->g(Z,\cdot).
4) D*_g becomes D* with this identification
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