  Kuramoto oscillator networks have the special property that their
trajectories are constrained to lie on the (at most) 3D orbits of the M\"obius
group acting on the state space $T^N$ (the $N$-fold torus). This result has
been used to explain the existence of the $N-3$ constants of motion discovered
by Watanabe and Strogatz for Kuramoto oscillator networks. In this work we
investigate geometric consequences of this M\"obius group action. The dynamics
of Kuramoto phase models can be further reduced to 2D reduced group orbits,
which have a natural geometry equivalent to the unit disk $\Delta$ with the
hyperbolic metric. We show that in this metric the original Kuramoto phase
model (with order parameter $Z_1$ equal to the centroid of the oscillator
configuration of points on the unit circle) is a gradient flow and the model
with order parameter $iZ_1$ (corresponding to cosine phase coupling) is a
completely integrable Hamiltonian flow. We give necessary and sufficient
conditions for general Kuramoto phase models to be gradient or Hamiltonian
flows in this metric. This allows us to identify several new infinite families
of hyperbolic gradient or Hamiltonian Kuramoto oscillator networks which
therefore have simple dynamics with respect to this geometry. We prove that for
the $Z_1$ model, a generic 2D reduced group orbit has a unique fixed point
corresponding to the hyperbolic barycenter of the oscillator configuration, and
therefore the dynamics are equivalent on different generic reduced group
orbits. This is not always the case for more general hyperbolic gradient or
Hamiltonian flows; the reduced group orbits may have multiple fixed points,
which also may bifurcate as the reduced group orbits vary.
