Computational Information Geometry Wonderland

Conjugate connections play a crucial role in information geometry. For dually flat statistical spaces, we have the exponential and mixture connections that are dual to each other. There have been several ways to define what we mean by a statistical manifold. The following paper summarizes the various approaches nicely.

Conjugate connections on statistical manifolds S. E. Stepanov, E. S. Stepanova and I. G. Shandra

It refers to a 1945 russian paper examining dual connections.

A. P. Norden, 
On Pairs of Conjugate Parallel Displacements in Multidimensional Spaces,
Dokl. Akad. Nauk SSSR 49 (9), 1345-1347 (1945).

Frank.