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The Cucker-Smale(CS) model is a velocity alignment model, and this model alsohas been generalized on general manifolds. We modify the CS model on manifoldsto get rid of a-priori condition on particles' positions and conditions oncommunication functions. Since the shortest geodesic is used to define aninteraction between two particles, if there exist two or more than two shortestgeodesics, then the system is not well-defined. In this paper, instead of usingthe shortest geodesic to define an interaction between two particles, we useall geodesics to define an interaction. From this assumption, we can relax thea-priori condition and conditions on communication functions. We also explainthe relationship between the suggested model and previous models. Finally, weprovide some emergent behaviors on some specific manifolds(e.g. flat torus,flat M\"{o}bius strip, and flat Klein bottle). From these results, we candiscuss the effect of the topology of the domain.

A velocity alignment model on quotient spaces of the Euclidean space