Generalization of the Fisher metric and Completely integrable system on Lie groups SO(2) and SO(3)

Abstract: In this article, we propose a generalization of the Fisher metric within the framework of Souriau's Lie group thermodynamics, focusing on the rotation groups $SO(2)$ and $SO(3)$. We characterize the effect of central 2-cocycles on the information geometry and integrability of gradient systems on the Lie groups SO(2) and SO(3) using the geometry of Jean-Marie Souriau. We also show how a cocycle can locally modify the Fisher metric on a coadjoint orbit, in explicit terms of brackets and central extensions on the Lie groups $SO(2)$ and $SO(3)$. The pioneering works of Fisher, S.I. Amari, Koszul Vinberg, and Souriau have each provided deep but often disjoint perspectives. Our strong motivation is to create a formal bridge between these theories, particularly by integrating Souriau's formalism on Lie group thermodynamics with the more classical structures of information geometry.