Riemannian polarization of multi-agent gradient flows

Stable polarization of multi-agent systems has been shown to exist over Rn and highly symmetric nonlinear spaces, especially the n-sphere Sn. Toward a more generalized setting without assuming linearity or symmetry, our previous work established the same type of emergent behavior over general hypersurfaces, subsuming the n-sphere case. In this paper, we discuss our ongoing work of extending our previous hypersurface results to study the stability of polarized equilibria of multi-agent gradient flows evolving on general Riemannian manifolds. The aim is to provide sufficient conditions in terms of the manifold geometry. Special attention is paid to two nonlinear manifolds of interest, the Stiefel manifold and the Grassmannian. While the polarization of the former share similar traits to that of the n-sphere, the latter is shown to have distinct polarization behaviors.