  In an attempt to find the dynamical foundations for $q$-entropies, we examine
the special case of Lagrangian/Hamiltonian systems of many degrees of freedom
whose statistical behavior is conjecturally described by the $q$-entropic
functionals. We follow the spirit of the canonical ensemble approach. We
consider the system under study as embedded in a far larger total system. We
explore some of the consequences that such an embedding has, if it is modelled
by a Riemannian submersion. We point out the significance in such a description
of the finite-dimensional Bakry-\'{E}mery Ricci tensor, as a local mesoscopic
invariant, for understanding the collective dynamical behavior of systems
described by the $q$-entropies.
