Refining LaSalle´s invariance principle

This work deals with the problem of locating the omega-limit set of a bounded solution of a given autonomous vector field on a Riemannian manifold. The derived results extend LaSalle´s invariance principle in such a way that the newly obtained conditions provide in certain situations a more refined statement about the location of the omega-limit set when the invariance principle is inconclusive. The derived conditions are for example useful for gradient-type vector fields and cascaded systems.