  This paper unifies the recent results on generalized Kuramoto Model
reductions. Lohe took a coupled system of $N$ bodies on $S^d$ governed by the
Kuramoto equations $\dot{x_i} = \Omega x_i + X - \langle x_i, X \rangle x_i$
and used the method of Watanabe and Strogatz to reduce this system to $d +
\frac{d(d-1)}{2}$ equations. Using a model of rigid rotations on a sphere as a
guide, we show that the reduction is described by a smooth path in the Lie
group of conformal transformations on the sphere, which is diffeomorphic to
$SO(d) \times D^d$. Seeing the reduction this way allows us to apply geometric
and topological reasoning in order to understand qualitative behavior of the
Kuramoto Model.

Dynamical Systems;Adaptation and Self-Organizing Systems