Within the context of non-linear variable-structure controlled systems, a general geometric characterization is introduced for the global existence of sliding motions on compact manifolds. As a necessary condition for the existence of global sliding motions, sign conditions are derived for the volume integrals of the divergence of the available feedback controlled vector fields. The ideal sliding dynamics are characterized in terms of a volume preserving flow or, equivalently, by a total zero-divergence smoothly controlled vector field. Several illustrative examples are given throughout the article.