This article presents a differential geometric approach for the design of variable structure controllers (VSC) leading to sliding regimes (Emelyanov 1967) in nonlinear smooth dynamical systems described in Rn. New properties, inherited from the geometry of the sliding manifold, are obtained in the controlled system when sliding conditions prevail. A sliding motion is obtained thanks to opportune switchings among the available structures in the feedback loop. This discontinuous control action is aimed at guaranteeing manifold reachability and invariance.