Emergent behaviors of relativistic flocks on Riemannian manifolds

We present a relativistic counterpart of the Cucker–Smale (CS) model on Riemannian manifolds (manifold RCS model in short) and study its collective behavior. For Euclidean space, the relativistic Cucker–Smale (RCS) model was introduced in Ha et al. (2020) via the method of a rational reduction from the relativistic gas mixture equations by assuming space-homogeneity, suitable ansatz for entropy and principle of subsystem. In this work, we extend the RCS model on Euclidean space to connected, complete and smooth Riemannian manifolds by replacing usual time derivative of velocity and relative velocity by suitable geometric quantities such as covariant derivative and parallel transport along length-minimizing geodesics. For the proposed model, we present a Lyapunov functional which decreases monotonically on generic manifolds, and show the emergence of weak velocity alignment on compact manifolds by using LaSalle's invariance principle. As concrete examples, we further analyze the RCS models on the unit sphere S^d and the hyperbolic space H^d. More precisely, we show that the RCS model on S^d exhibits a dichotomy in asymptotic spatial patterns, and provide a sufficient framework leading to the velocity alignment of RCS particles in H^d. For the hyperbolic space H^d, we also rigorously justify smooth transition from the RCS model to the CS model in any finite time interval, as speed of light tends to infinity.