  Multi-agent systems are known to exhibit stable emergent behaviors, including
polarization, over $\mathbb{R}^n$ or highly symmetric nonlinear spaces. In this
article, we eschew linearity and symmetry of the underlying spaces, and study
the stability of polarized equilibria of multi-agent gradient flows evolving on
general hypermanifolds. The agents attract or repel each other according to the
partition of the communication graph that is connected but otherwise arbitrary.
The manifolds are outfitted with geometric features styled ``dimples'' and
``pimples'' that characterize the absence of flatness. The signs of inter-agent
couplings together with these geometric features give rise to stable
polarization under various sufficient conditions. We propose tangible
interpretation of the system in the context of opinion dynamics, and highlight
throughout the text its versatility in modeling various aspects of the
polarization phenomenon.
