In this paper, we clarify several issues concerning the abstract geometrical formulation of thermodynamics on non compact symmetric spaces $\mathrm{U/H}$ that are the mathematical model of hidden layers in the new paradigm of Cartan Neural Networks. We introduce a distinction between the generalized thermodynamics associated with Dynamical Systems and the challenging proposal of Gibbs probability distributions on $\mathrm{U/H}$ provided by generalized thermodynamics {à} la Souriau. Main result is the proof that $\mathrm{U/H}$.s supporting Gibbs distributions are only the Kähler ones. For the latter, we solve the problem of determining the space of temperatures, namely of Lie algebra elements for which the partition function converges. The space of generalized temperatures is the orbit under the adjoint action of $\mathrm{U}$ of a positivity domain in the Cartan subalgebra $C_c\subset\mathbb{H}$ of the maximal compact subalgebra $\mathbb{H}\subset\mathbb{U}$. We illustrate how our explicit constructions for the Poincaré and Siegel planes might be extended to the whole class of Calabi-Vesentini manifolds utilizing Paint Group symmetry. Furthermore we claim that Rao's, Chentsov's, Amari's Information Geometry and the thermodynamical geometry of Ruppeiner and Lychagin are the very same thing. The most important property of the Gibbs probability distributions provided by the here introduced setup is their covariance with respect to the action of the full group of symmetries $\mathrm{U}$. The partition function is invariant against $\mathrm{U}$ transformations and the set of its arguments, namely the generalized temperatures, can be always reduced to a minimal set whose cardinality is equal to the rank of the compact denominator group $\mathrm{H}\subset \mathrm{U}$.

翻译：本文澄清了关于非紧对称空间$\mathrm{U/H}$上热力学抽象几何公式化的若干问题，这类空间是Cartan神经网络新范式中隐藏层的数学模型。我们区分了与动力系统相关的广义热力学，以及由Souriau广义热力学在$\mathrm{U/H}$上提出的Gibbs概率分布这一具有挑战性的方案。主要结果是证明了仅Kähler型$\mathrm{U/H}$支持Gibbs分布。对于后者，我们解决了温度空间的确定问题，即李代数中使配分函数收敛的元素构成的集合。广义温度空间是紧致子代数$\mathbb{H}\subset\mathbb{U}$的Cartan子代数$C_c\subset\mathbb{H}$中正性域在$\mathrm{U}$的伴随作用下的轨道。我们展示了如何利用Paint群对称性将庞加莱平面和Siegel平面的显式构造推广到整个Calabi-Vesentini流形类。此外，我们断言Rao、Chentsov和Amari的信息几何与Ruppeiner及Lychagin的热力学几何本质上是相同的。本文框架提供的Gibbs概率分布最重要的性质是它们关于全对称群$\mathrm{U}$作用的协变性。配分函数在$\mathrm{U}$变换下保持不变，且其自变量集（即广义温度）始终可约化为最小集，其基数等于紧致分母群$\mathrm{H}\subset \mathrm{U}$的秩。