Connections with torsion, parallel spinors and geometry of Spin(7) manifolds

Abstract

We show that on every Spin(7)-manifoldthere always exists a unique linear connection with totally skew-symmetric torsion preserving a nontrivial spinor andthe Spin(7) structure. We express its torsion andthe Riemannian scalar cur- vature 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 ˆ A-genus andobtain a linear relation between Betti numbers of a compact Spin(7) manifoldwhich is locally but not globally conformally equivalent to a space with closedfund amental 4-form. A general solution to the Killing spinor equations is presented.