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We investigate the long-time behaviour of solutions to a nonlocal partialdifferential equation on smooth Riemannian manifolds of bounded sectionalcurvature. The equation models self-collective behaviour with intrinsicinteractions that are modelled by an interaction potential. We considerattractive interaction potentials and establish sufficient conditions for aconsensus state to form asymptotically. In addition, we quantify the approachto consensus, by deriving a convergence rate for the diameter of the solution'ssupport. The analytical results are supported by numerical simulations for theequation set up on the rotation group.

Long-time behaviour of interaction models on Riemannian manifolds with bounded curvature