Interplay of geometric constraint and bonding force in the emergent behaviors of relativistic Cucker-Smale flocks

Abstract: We present the relativistic analogue of the Cucker-Smale model with a bonding force on Riemannian manifold, and study its emergent dynamics. The Cucker-Smale model serves a prototype example of mechanical flocking models, and it has been extensively studied from various points of view. Recently, the authors studied collision avoidance and asymptotic flocking of the Cucker-Smale model with a bonding force on the Euclidean space. In this paper, we provide an analytical framework for collision avoidance and asymptotic flocking of the proposed model on Riemannian manifolds. Our analytical framework is explicitly formulated in terms of system parameters, initial data and the injectivity radius of the ambient manifold, and we study how the geometric information of an ambient manifold can affect the flocking dynamics.