2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/167063We obtain geometric estimates for the first eigenvalue and the fundamental tone of the p-laplacian on manifolds in terms of admissible vector fields. Also, we defined a new spectral invariant and we show its relation with the geometry of the manifold.Differential Geometry58C40, 53C2text