2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/78272We present the details of our embedding proof of the Atiyah-Patodi-Singer index theorem for Dirac operators on manifolds with boundary.Differential GeometryAnalysis of PDEs58GxxReal embeddings and the Atiyah-Patodi-Singer index theorem for Dirac operatorstext