2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/169391We construct a natural co-Riemannian structure on the manifold of smooth loops in a Riemannian manifold. We show that the smooth loop space of a string manifold is a per-Hilbert-Schmidt locally equivalent co-spin manifold and thus admits a Dirac operator.30 pages. Together with arXiv:0809.3401 this replaces math.DG/0505077Differential Geometry58B20 (Primary); 58D15, 53C15 (Secondary)text