2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/99956We prove the Focal Index Lemma and the Rauch and Berger comparison theorems on a weak Riemannian Hilbert manifold with a smooth Levi-Civita connection and we apply these results to the free loop spaces of a compact manifold with the L^2 metrics12 pagesDifferential GeometryPrimary 58B20 secondary 58D15, 22E65text