Metric Spaces

– Maurice Fréchet a introduit en 1939, la borne inférieure de la variance de tout estimateur statistique, donnée par l'inverse de la matrice de Fisher, cette dernière définissant la métrique de la « Géométrie de l'Information ». Le mathématicien Jean-Louis Koszul a montré que cette métrique pouvait être définie de façon plus générale comme le hessien de l'Entropie (transformée de Legendre du logarithme de la fonction caractéristique). De façon analogue, le physicien Roger Balian a élaboré une métrique de Fisher quantique comme le hessien de l'Entropie de von Neumann. Enfin, le physicien Jean-Marie Souriau a donné une nouvelle version covariante de cette métrique de Fisher pour les groupes dynamiques, où elle apparaît comme une capacité thermique géométrique via le cocycle symplectique lié au groupe et une température géométrique. Abstract-Maurice Fréchet introduced in 1939, the lower bound of any statistical estimator variance, given by the inverse of the Fisher matrix, the latter defining the " Information Geometry" metric. The mathematician Jean-Louis Koszul showed that this metric could be defined in a more general way as the hessian of the Entropy (Legendre transform of the characteristic function logarithm). In a similar way, the physicist Roger Balian developed a quantum Fisher metric as the hessian of the von Neumann Entropy. Finally, the physicist, Jean-Marie Souriau gave a new covariant version of this Fisher metric for the dynamic groups, where it appears as a geometrical heat capacity, via the symplectic cocycle linked to the group and a geometrical temperature.

In this paper we study on contribution of fixed point theorem in Metric spaces and Quasi Metric spaces. Definition: 1 (Metric Space) Let X be a non-empty set-A function XxX →R (the set of reals) such that p:XxX→R is called a metric or distance function (if ad only if) p satisfies the following conditions. (i) p (x,y) ≥ 0 for all x, y x (ii) p (x,y)= 0 if x=y (iii) p (x,y) = p (y,x) for all x,y x (iv) p (x,y) ≥ p (x,z)+ p (z ,y) for all x,y,z X If p is a metric for X, then the pair (X, p) is called a metric space. Definition: 2 (Cauchy sequence) Let (X,p) be a metric space .Then a sequence { xn} of points of X is said to be a Cauchy sequence if for each >0,there exists a positive integer n0 such that m,nn0 implies p(xm,xn) < . Definition: 3 (Completeness): A metric space(X,p) is said to be complete if every Cauchy sequence in X converges to point of X. Definition: 4 (Contraction mapping) Let (X, p) be a complete metric space. A mapping T:XX is said to be a contraction mapping if there exists a real number  with 0 << 1 such that, p{T(x) ,T(y)})  p (x,y) <p(x,y)  x, y,  X i.e In a contraction mapping, the distance between the images of any two points is less than the distance between the points.

We introduce the Symplectic Structure of Information Geometry based on Souriau's Lie Group Thermodynamics model, with a covariant definition of Gibbs equilibrium via invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat, and momentum as pure geometrical objects. Using Geometric (Planck) Temperature of Souriau model and Symplectic cocycle notion, the Fisher metric is identified as a Souriau Geometric Heat Capacity. Finally, in the framework of Lie Group Thermodynamics, an Euler-Poincaré equation is elaborated with respect to thermodynamic variables. We conclude with a new variational principal for thermodynamics built through an invariant Poincaré-Cartan-Souriau integral.

The François Massieu 1869 idea to derive some mechanical and thermal properties of physical systems from " Characteristic Functions " , was developed by Gibbs and Duhem in thermodynamics with the concept of potentials, and introduced by Poincaré in probability. This paper deals with generalization of this Characteristic Function concept by Jean-Louis Koszul in Mathematics and by Jean-Marie Souriau in Statistical Physics. The Koszul-Vinberg Characteristic Function (KVCF) on convex cones will be presented as cornerstone of " Information Geometry " theory, defining Koszul Entropy as Legendre transform of minus the logarithm of KVCF, and Fisher Information Metrics as hessian of these dual functions, invariant by their automorphisms. In parallel, Souriau has extended the Characteristic Function in Statistical Physics looking for other kinds of invariances through co-adjoint action of a group on its momentum space, defining physical observables like energy, heat and momentum as pure geometrical objects. In covariant Souriau model, Gibbs equilibriums states are indexed by a geometric parameter, the Geometric (Planck) Temperature, with values in the Lie algebra of the dynamical Galileo/Poincaré groups, interpreted as a space-time vector, giving to the metric tensor a null Lie derivative. Fisher Information metric appears as the opposite of the derivative of Mean " Moment map " by geometric temperature, equivalent to a Geometric Capacity or Specific Heat. These elements has been developed by author in [10][11]. KOSZUL CHARACTERISTIC FUNCTION/ENTROPY BY LEGENDRE DUALITY We define Koszul-Vinberg Hessian metric on convex sharp cone, and observe that the Fisher information metric of Information Geometry coincides with the canonical Koszul Hessian metric (given by Koszul forms). We also observe, by Legendre duality (Legendre transform of minus Koszul characteristic function logarithm), that we are able to introduce a Koszul Entropy, that plays the role of generalized Shannon Entropy. Koszul [1][2][3][4] and E. Vinberg have introduced an affinely invariant hessian metric on a sharp convex cone Ω  through its characteristic function ψ. In the following, Ω is a sharp open convex cone in a vector space E of finite dimension on R (a convex cone is sharp if it does not contain any full straight line). In dual space *