  Boltzmann's principle S(E,N,V...)=ln W(E,N,V...) allows the interpretation of
Statistical Mechanics of a closed system as Pseudo-Riemannian geometry in the
space of the conserved parameters E,N,V... (the conserved mechanical parameters
in the language of Ruppeiner) without invoking the thermodynamic limit. The
topology is controlled by the curvature of S(E,N,V...). The most interesting
region is the region of (wrong) positive maximum curvature, the region of
phase-separation. This is demonstrated among others for the equilibrium of a
typical non-extensive system, a self-gravitating and rotating cloud in a
spherical container at various energies and angular-momenta. A rich variety of
realistic configurations, as single stars, multi-star systems, rings and
finally gas, are obtained as equilibrium microcanonical phases. The global
phase diagram, the topology of the curvature, as function of energy and
angular-momentum is presented. No exotic form of thermodynamics like Tsallis
non-extensive one is necessary. It is further shown that a finite (even
mesoscopic) system approaches equilibrium with a change of its entropy Delta
S\ge 0 (Second Law) even when its Poincarre recurrence time is not large.
