2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/62690We show that on every Spin(7) manifold there always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor and the Spin(7) structure. We express its torsion and the Riemannian scalar curvature in terms of the fundamental 4-form. We present an explicit formula for the Riemannian covariant derivative of the fundamental 4-form in terms of its exterior differential. We show the vanishing of the (\hat)-A genus and obtain a linear relation between Betti numbers of a compact Spin(7) manifolds which are locally but not globally conformally equivalent to a space with closed fundamental 4-form. A general solution to the Killing spinor equations is presentedLaTeX, 15 pages, references added, some typos corrected, final version to appear in Math. Res. LettDifferential GeometryHigh Energy Physics - Theory53C25text