  Certain consensus seeking multi-agent systems can be formulated as gradient
descent flows of a disagreement function. We study how known pathologies of
gradient descent flows in Euclidean spaces carry over to consensus seeking
systems that evolve on nonlinear manifolds. In particular, we show that the
norms of agent states can diverge to infinity, but this will not happen if the
manifold is the boundary of a convex set. Moreover, the system can be
initialized arbitrarily close to consensus without converging to it, but this
will not happen if the manifold is analytic. For analytic manifolds, consensus
is asymptotically stable. This last result summarizes a number of previous
findings in the literature on generalizations of the well-known Kuramoto model
to high-dimensional manifolds.
