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

TL;DR

This paper investigates the curvature identities and Ricci soliton structures on compact $Spin(7)$-manifolds with torsion, establishing conditions for Ricci flatness, parallel torsion, and generalized Ricci solitons.

Contribution

It provides new curvature identities and characterizations of Ricci flatness and soliton structures on $Spin(7)$-manifolds with torsion, linking geometric properties to torsion and Ricci tensor conditions.

Findings

Curvature $R$ vanishes Ricci tensor iff torsion is parallel.

Riemannian Bianchi identity holds iff torsion is parallel.

Closed torsion implies Ricci flatness under constant torsion norm or scalar curvature.

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.