Continuum modeling and analysis of collective motion on Riemannian manifolds

In this study, we explore a sufficient framework for the continuum Cucker–Smale model. To this end, we first formulate a continuum version of the Cucker–Smale model in Euclidean space, represented as an integro–differential equation obtained by taking the continuum limit of the standard Cucker–Smale model. Subsequently, we establish fundamental estimates for this model and present sufficient conditions for the emergence of asymptotic flocking. Furthermore, we extend the model to a Riemannian manifold setting and develop a sufficient framework for describing collective dynamics. In particular, we investigate collective behavior on specific manifolds, namely the $ d $-dimensional sphere $ \mathbb{S}^d $ and the $ d $-dimensional hyperbolic space $ \mathbb{H}^d $.