2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/67130In this paper we will investigate the global properties of complete Hilbert manifolds with upper and lower bounded sectional curvature. We shall prove the Focal Index Lemma that we will allow us to extend some classical results of finite dimensional Riemannian geometry such as Rauch and Berger Theorems and the Topogonov Theorem in the class of manifolds on which the Hopf-Rinow Theorem holds.27 pages, no figuresDifferential Geometry2000 Mathematical classification Primary 58B20; Secondary 53C21text