Amari Functors and Dynamics in Gauge Structures

We deal with finite dimensional differentiable manifolds. All items are concerned with are differentiable as well. The class of differentiability is C∞. A metric structure in a vector bundle E is a constant rank symmetric bilinear vector bundle homomorphism of E × E in the trivial bundle line bundle. We address the question whether a given gauge structure in E is metric. That is the main concerns. We use generalized Amari functors of the information geometry for introducing two index functions defined in the moduli space of gauge structures in E. Beside we introduce a differential equation whose analysis allows to link the new index functions just mentioned with the main concerns. We sketch applications in the differential geometry theory of statistics. Reader interested in a former forum on the question whether a giving connection is metric are referred to appendix.