The 3-dimensional gravitational N-body problem, N ≥ 2, in spaces of constant Gaussian curvature kappa ≠ 0, i.e. on spheres S3k, for kappa > 0, and on hyperbolic manifolds H3k, for kappa < 0, is considered. In the 3-dimensional curved N-body problem, the new concept of rotopulsating orbits is defined. This type of solution is used when the bodies rotate and change size during the motion. Considering the possibility of having these bodies in spaces of positive or negative curvature, it is feasible to use the following classification: positive elliptic, positive elliptic-elliptic, negative elliptic, negative hyperbolic, and negative elliptic-hyperbolic. The necessary and sufficient criteria for the existence of rotopulsators are provided. Results will be obtained that describe their qualitative behaviour, which will then be applied to find examples for each type of rotopulsating orbits.
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