  We consider the N-body problem in spaces of constant curvature and study its
rotopulsators, i.e.\ solutions for which the configuration of the bodies
rotates and changes size during the motion. Rotopulsators fall naturally into
five groups: positive elliptic, positive elliptic-elliptic, negative elliptic,
negative hyperbolic, and negative elliptic-hyperbolic, depending on the nature
and number of their rotations and on whether they occur in spaces of positive
or negative curvature. After obtaining existence criteria for each type of
rotopulsator, we derive their conservation laws. We further deal with the
existence and uniqueness of some classes of rotopulsators in the 2- and 3-body
case and prove two general results about the qualitative behaviour of
rotopulsators. More precisely, for positive curvature we show that there is no
foliation of the 3-sphere with Clifford tori such that the motion of each body
is confined to some Clifford torus. For negative curvature, a similar result is
proved relative to foliations of the hyperbolic 3-sphere with hyperbolic
cylinders.
