  The Kuramoto model of coupled phase oscillators is often used to describe
synchronization phenomena in nature. Some applications, e.g., quantum
synchronization and rigid-body attitude synchronization, involve
high-dimensional Kuramoto models where each oscillator lives on the n-sphere or
SO(n). These manifolds are special cases of the compact, real Stiefel manifold
St(p,n). Using tools from optimization and control theory, we prove that the
generalized Kuramoto model on St(p,n) converges to a synchronized state for any
connected graph and from almost all initial conditions provided (p,n) satisfies
p<=2n/3-1 and all oscillator frequencies are equal. This result could not have
been predicted based on knowledge of the Kuramoto model in complex networks
over the circle. In that case, almost global synchronization is graph
dependent; it applies if the network is acyclic or sufficiently dense. This
paper hence identifies a property that distinguishes many high-dimensional
generalizations of the Kuramoto models from the original model.
