Geometry of hypersurfaces defined by the relation which generalizes classical formula for free energy in terms of microstates is studied. Induced metric, Riemann curvature tensor, Gauss-Kronecker curvature and associated entropy are calculated. Speci
al class of ideal statistical hypersurfaces is analyzed in details. Non-ideal hypersurfaces and their singularities similar to those of the phase transitions are considered. Tropical limit of statistical hypersurfaces and double scaling tropical limit are discussed too.
Interrelation between Thoms catastrophes and differential equations revisited. It is shown that versal deformations of critical points for singularities of A,D,E type are described by the systems of Hamilton-Jacobi type equations. For particular nonv
ersal unfoldings the corresponding equations are equivalent to the integrable two-component hydrodynamic type systems like classical shallow water equation and dispersionless Toda system and others. Pecularity of such integrable systems is that the generating functions for corresponding hierarchies, which obey Euler-Poisson-Darboux equation, contain information about normal forms of higher order and higher corank singularities.
