  The Kuramoto model was recently extended to arbitrary dimensions by
reinterpreting the oscillators as particles moving on the surface of unit
spheres in a D-dimensional space. Each particle is then represented by a
D-dimensional unit vector. For $D=2$ the particles move on the unit circle and
the vectors can be described by a single phase, recovering the original
Kuramoto model. This multidimensional description can be further extended by
promoting the coupling constant between the particles to a matrix that acts on
the unit vectors, representing a type of generalized frustration. In a recent
paper we have analyzed in detail the role of the coupling matrix for $D=2$.
Here we extend this analysis to arbitrary dimensions, presenting a study of
synchronous states and their stability. We show that when the natural
frequencies of the particles are set to zero, the system converges either to a
stationary synchronized state with well defined phase, or to an effective
two-dimensional dynamics, where the synchronized particles rotate on the
sphere. The stability of these states depend on the eigenvalues and
eigenvectors of the coupling matrix. When the natural frequencies are not zero,
synchronization depends on whether $D$ is even or odd. In even dimensions the
transition to synchronization is continuous and rotating states are replaced by
active states, where the order parameter rotates while its module oscillates.
If $D$ is odd the phase transition is discontinuous and active states are
suppressed, occurring only for a restricted class of coupling matrices.
