  We study a system of $N$ interacting particles moving on the unit sphere in
$d$-dimensional space. The particles are self-propelled and coupled all to all,
and their motion is heavily overdamped. For $d=2$, the system reduces to the
classic Kuramoto model of coupled oscillators; for $d=3$, it has been proposed
to describe the orientation dynamics of swarms of drones or other entities
moving about in three-dimensional space. Here we use group theory to explain
the recent discovery that the model shows low-dimensional dynamics for all $N
\ge 3$, and to clarify why it admits the analog of the Ott-Antonsen ansatz in
the continuum limit $N \rightarrow \infty$. The underlying reason is that the
system is intimately connected to the natural hyperbolic geometry on the unit
ball $B^d$. In this geometry, the isometries form a Lie group consisting of
higher-dimensional generalizations of the M\"obius transformations used in
complex analysis. Once these connections are realized, the reduced dynamics and
the generalized Ott-Antonsen ansatz follow immediately. This framework also
reveals the seamless connection between the finite and infinite-$N$ cases.
Finally, we show that special forms of coupling yield gradient dynamics with
respect to the hyperbolic metric, and use that fact to obtain global stability
results about convergence to the synchronized state.
