The Riemannian curvature identities for the torsion connection on $Spin(7)$-manifold and generalized Ricci solitons

Abstract

It is shown that on compact $Spin(7)$--manifold with exterior derivative of the Lee form lying in the Lie algebra $spin(7)$ the curvature $R$ of the $Spin(7)$--torsion connection $R\in S^2\Lambda^2$ with vanishing Ricci tensor if and only if the $3$-form torsion is parallel with respect to the Levi-Civita connection. It is also proved that $R$ satisfies the Riemannian first Bianchi identity exactly when the $3$-form torsion is parallel with respect to the Levi-Civita and to the $Spin(7)$--torsion connections simultaneously. Precise conditions for a compact $Spin(7)$--manifold to has closed torsion are given in terms of the Ricci tensor of the $Spin(7)$--torsion connection. It is shown that a compact $Spin(7)$--manifold with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature is constant. It is proved that any compact $Spin(7)$--manifold with closed torsion 3-form is a generalized gradient Ricci soliton and this is equivalent to a certain vector field to be parallel with respect to the torsion connection. In particular, this vector field preserves the $Spin(7)$--structure.