The mean-field limit of the Cucker-Smale model on complete Riemannian manifolds

We study the mean-field limit of the Cucker-Smale (C-S) model for flocking on complete smooth Riemannian manifolds. For this, we first formally derive the kinetic manifold C-S model on manifolds using the BBGKY hierarchy and derive several a priori estimates on emergent dynamics. Then, we present a rigorous mean-field limit from the particle model to the corresponding kinetic model by using the generalized particle-in-cell method. As a byproduct of our rigorous mean-field limit estimate, we also establish a global existence of a measure-valued solution for the derived kinetic model. Compared to the corresponding results on $\mathbb {R}^d$, our procedure requires additional assumption on holonomy and proper a priori bound on the derivative of parallel transports. As a concrete example, we verify that hyperbolic space $\mathbb {H}^d$ satisfies our proposed standing assumptions.