Title: Amari Functors and Dynamics in Gauge Structures,
Abstract: We deal with finite dimensional differentiable manifolds. All items are
concerned with are differentiable as well. The class of differentiability is
$C^\infty$. A metric structure in a vector bundle $E$ is a constant rank
symmetric bilinear vector bundle homomorphism of $E\times E$ 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.