  I propose a model of mutually interacting particles on an M-dimensional unit
sphere. I derive the dynamics of the particles by extending the dynamics of the
Kuramoto-Sakaguchi model. The dynamics include a natural-frequency matrix,
which determines the motion of a particle with no external force, and an
external force vector. The position (state variable) of a particle at a given
time is obtained by the projection transformation of the initial position of
the particle. The same projection transformation gives the position of the
particles with the same natural-frequency matrix. I show that the motion of the
centre of mass of an infinite number of heterogeneous particles whose
natural-frequency matrices are obtained from a class of multivariate Lorentz
distribution is given by an M-dimensional ordinary differential equation in
closed form. This result is an extension of the Ott-Antonsen theory.
