  Consensus on nonlinear spaces is of use in many control applications. This
paper proposes a gradient descent flow algorithm for consensus on
hypersurfaces. We show that if an inequality holds, then the system converges
for almost all initial conditions and all connected graphs. The inequality
involves the hypersurface Gauss map and the gradient and Hessian of the
implicit equation. Moreover, for the inequality to hold, it is necessary that
the manifold is the boundary of a convex set. The literature already contains
an algorithm for consensus on hypersurfaces. That algorithm on any ellipsoid is
equivalent to our algorithm on the unit sphere. In particular, that algorithm
achieves almost global synchronization on ellipsoids. These findings suggest
that strong convergence results for consensus seeking gradient descent flows
may be established on manifolds that are the boundaries of convex sets.
