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 ${\mathbb{S}}^d$ and the hyperbolic space ${\mathbb{H}}^d$. More precisely, we show that the RCS model on ${\mathbb{S}}^d$ exhibits a dichotomy in asymptotic spatial patterns, and provide a sufficient framework leading to the velocity alignment of RCS particles in ${\mathbb{H}}^d$. For the hyperbolic space ${\mathbb{H}}^d$, 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.

我们在黎曼流形（简称流形 RCS 模型）上提出了 Cucker-Smale (CS) 模型的相对论对应物，并研究了它的集体行为。对于欧几里得空间，相对论 Cucker-Smale(RCS) 模型是在 Ha 等人中引入的。(2020) 通过假设空间均匀性、适用于熵的 ansatz 和子系统原理，从相对论气体混合物方程中合理归约的方法。在这项工作中，我们通过用合适的几何量（例如协变导数和沿长度最小的测地线的平行传输）替换速度和相对速度的通常时间导数，将欧几里得空间上的 RCS 模型扩展到连接的、完整的和光滑的黎曼流形。对于所提出的模型，我们提出了一个 Lyapunov 泛函，它在通用流形上单调递减，并通过使用 LaSalle 不变性原理展示了紧凑流形上弱速度对齐的出现。作为具体例子，我们进一步分析单位球体上的RCS模型${\mathbb{秒}}^d$ 和双曲空间 ${\mathbb{H}}^d$. 更准确地说，我们表明 RCS 模型在${\mathbb{秒}}^d$ 在渐近空间模式中表现出二分法，并提供了一个足够的框架，导致 RCS 粒子在 ${\mathbb{H}}^d$. 对于双曲空间${\mathbb{H}}^d$，我们还严格证明在任何有限时间间隔内从 RCS 模型到 CS 模型的平滑过渡是合理的，因为光速趋于无穷大。