2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/122908For a compact spin manifold $M$ isometrically embedded into Euclidean space, we derive the extrinsic estimates from above and below for eigenvalues of the Dirac operators, which depend on the second fundamental form of the embedding. We also show the bounds of the ratio of the eigenvalues.12pages,submitted to "Math.Z.", Jun 2006Differential Geometry53C27,34L15, 35P15text