In this paper we clarify the relation between Geometric Thermodynamics and Information Geometry based on the Fisher matrix. On the macroscopic odd-dimensional contact manifold of thermodynamic variables, we introduce for the first time a metric, whose pull-back on the isoentropic symplectic submanifolds transverse to the Reeb field is Kählerian. The pull-back of such metric on equilibrium states, that are lagrangian submanifolds, is the Fisher Hessian. Then we consider the Souriau-like Thermodynamics that uses Calabi-Vesentini (CV) manifolds as Kaehlerian microscopic event manifolds and the Killing moment maps as observable functions. A systematic use of the theory of compact abelian structures and the setup of Special Kähler Geometry in which CV manifolds are encoded allows us to perform the explicit integration defining the partition function for any entry in the CV Tits Satake universality class. The additional actions completing the abelian structure are non linear Casimir functions of the Killing moment-maps and suggest a generalization of Souriau thermodynamics that partially breaks the isometry group symmetry by means of the non vanishing mean values of the Casimir functions in a manner similar to the spontaneous magnetization in ferromagnetism. Our new exact Gibbs distributions provide the analogue for Cartan Neural Networks of the Gaussian probability distributions in flat space used in conventional Machine Learning.

翻译：本文阐明了基于Fisher矩阵的几何热力学与信息几何之间的关系。在由热力学变量构成的宏观奇数维接触流形上，我们首次引入了一个度规，其在垂直于Reeb场的等熵辛子流形上的拉回是Kähler的。该度规在作为拉格朗日子流形的平衡态上的拉回即为Fisher Hessian。随后，我们考虑了类似Souriau的热力学，该理论采用Calabi-Vesentini (CV)流形作为Kähler微观事件流形，并利用Killing矩映射作为可观测量函数。通过系统运用紧致阿贝尔结构理论以及嵌入CV流形的特殊Kähler几何框架，我们能够对CV Tits Satake万有类中的任意条目进行显式积分，从而定义配分函数。完成阿贝尔结构的附加作用量是Killing矩映射的非线性Casimir函数，这暗示了Souriau热力学的一种推广：通过Casimir函数的非零平均值部分破缺等度量群对称性，其方式类似于铁磁现象中的自发磁化。我们得到的新精确Gibbs分布为Cartan神经网络提供了类似于传统机器学习中平坦空间高斯概率分布的对应物。