The Riemannian curvature identities for the torsion connection on Spin(7)${\rm Spin}(7)$—Manifold and generalized Ricci solitons

It is shown that on compact Spin(7)${\rm Spin}(7)$‐manifold with exterior derivative of the Lee form lying in the Lie algebra Spin(7)${\rm Spin}(7)$ the curvature R$R$ of the Spin(7)${\rm Spin}(7)$–torsion connection R∈S2Λ2$R\in S^2\Lambda ^2$ 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$R$ 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)${\rm Spin}(7)$‐torsion connections simultaneously. Precise conditions for a compact Spin(7)${\rm Spin}(7)$‐manifold to has closed torsion are given in terms of the Ricci tensor of the Spin(7)${\rm Spin}(7)$‐torsion connection. It is shown that a compact Spin(7)${\rm Spin}(7)$‐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)${\rm Spin}(7)$‐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)${\rm Spin}(7)$‐structure.