  Microcanonical description is characterized by the presence of an internal
symmetry closely related with the dynamical origin of this ensemble: the
reparametrization invariance. Such symmetry possibilities the development of a
non Riemannian geometric formulation within the microcanonical description of
an isolated system, which leads to an unexpected generalization of the Gibbs
canonical ensemble and the classical fluctuation theory for the open systems
(where the inverse temperature and the total energy E behave as complementary
thermodynamical quantities), the improvement of Monte Carlo simulations based
on the canonical ensemble, as well as a reconsideration of any classification
scheme of the phase transitions based on the concavity of the microcanonical
entropy.
