Opinion Dynamics on a General Compact Riemannian Manifold

This work formulates the problem of defining a model for opinion dynamics on a general compact Riemannian manifold. Two approaches to modeling opinions on a manifold are explored. The first defines the distance between two points using the projection in the ambient Euclidean space. The second approach defines the distance as the length of the geodesic between two agents. Our analysis focuses on features such as equilibria, the long term behavior, and the energy of the system, as well as the interactions between agents that lead to these features. Simulations for specific manifolds, \begin{document} $\mathbb{S}^1, \mathbb{S}^2,$ \end{document} and \begin{document} $\mathbb{T}^2$ \end{document} , accompany the analysis. Trajectories given by opinion dynamics may resemble \begin{document} $n-$ \end{document} body Choreography and are called "social choreography". Conditions leading to various types of social choreography are investigated in \begin{document} $\mathbb{R}^2$ \end{document} .