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Multi-agent systems on nonlinear spaces sometimes fail to synchronize. Thisis usually attributed to the initial configuration of the agents being toospread out, the graph topology having certain undesired symmetries, or both.Besides nonlinearity, the role played by the geometry and topology of thenonlinear space is often overlooked. This paper concerns two gradient descentflows of quadratic disagreement functions on general Riemannian manifolds. Onesystem is intrinsic while the other is extrinsic. We derive necessaryconditions for the agents to synchronize from almost all initial conditionswhen the graph used to model the network is connected. If a Riemannian manifoldcontains a closed curve of locally minimum length, then there is a connectedgraph and a dense set of initial conditions from which the intrinsic systemfails to synchronize. The extrinsic system fails to synchronize if the manifoldis multiply connected. The extrinsic system appears in the Kuramoto model on$\smash{\mathcal{S}^1}$, rigid-body attitude synchronization on$\mathsf{SO}(3)$, the Lohe model of quantum synchronization on the $n$-sphere,and the Lohe model on $\mathsf{U}(n)$. Except for the Lohe model on the$n$-sphere where $n\in\mathbb{N}\backslash\{1\}$, there are dense sets ofinitial conditions on which these systems fail to synchronize. The reason forthis difference is that the $n$-sphere is simply connected for all$n\in\mathbb{N}\backslash\{1\}$ whereas the other manifolds are multiplyconnected.

A Geometric Obstruction to Almost Global Synchronization on Riemannian Manifolds