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：本文旨在阐明非紧对称空间 $\mathrm{U/H}$ 上热力学抽象几何表述中的若干问题，该空间是卡坦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和索里奥广义热力学在 $\mathrm{U/H}$ 上提出的吉布斯概率分布这一挑战性方案。主要结果证明了 $\mathrm{U/H}$ 中支持吉布斯分布的空间仅为凯勒型空间。对于后者，我们解决了确定温度空间的问题，即划分函数收敛的李代数元素集合。广义温度空间是极大紧子代数 $\mathbb{H}\subset\mathbb{U}$ 的卡坦子代数 $C_c\subset\mathbb{H}$ 中正性域在 $\mathrm{U}$ 的伴随作用下的轨道。我们阐述了如何利用Paint群对称性，将庞加莱平面和西格尔平面上的显式构造推广至整个卡拉比-维森蒂尼流形类。此外，我们主张Rao、Chentsov、Amari的信息几何与Ruppeiner、Lychagin的热力学几何本质相同。本文所引入框架提供的吉布斯概率分布的最重要性质是其关于完整对称群 $\mathrm{U}$ 作用的协变性。划分函数在 $\mathrm{U}$ 变换下保持不变，其自变量集合（即广义温度）总可约化为最小集合，该集合的基数等于紧分母群 $\mathrm{H}\subset \mathrm{U}$ 的秩。