Connections with torsion, parallel spinors and geometry of Spin(7) manifolds
The existence of a parallel spinor imposes restrictions on the holonomy group. In the case of Levi-Civita connections the possible Riemannian holonomy groups are \(\text{SU}(n)\), \(\text{Sp}(n)\), \(G_2\) and \(\text{Spin}(7)\). In this paper, the author studies the existence of parallel spinors with respect to a metric connection with skew-symmetric torsion in manifolds of dimension \(8\). He proves that such a manifold must be a \(\text{Spin}(7)\)-manifold. He shows also the vanishing of the \(A\)-genus and obtains a linear relation between Betti numbers of a compact \(\text{Spin}(7)\)-manifold which is locally (but not globally) conformally equivalent to a space with closed fundamental 4-form. Moreover, a general solution to the Killing spinor equations is also presented.
scientific article