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