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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.