A velocity alignment model on quotient spaces of the Euclidean space

The Cucker–Smale (CS) model is a velocity alignment model, and this also has been generalized on Riemannian manifolds. We modify the CS manifolds to get rid of a-priori condition particles' positions conditions communication functions. Since shortest geodesic used define an interaction between two particles, if there exist or more than geodesics, then system not well-defined. In paper, instead using we use all geodesics interaction. From assumption, can relax explain relationship suggested previous models. Finally, provide some emergent behaviors specific (e.g. flat torus, Möbius strip, Klein bottle). these results, discuss how topology domain effects