The Riemannian Bianchi identities of metric connections with skew torsion and generalized Ricci solitons

TL;DR

This paper explores the curvature properties of metric connections with skew torsion, establishing conditions under which scalar curvature and Ricci solitons are characterized, and deriving identities related to Riemannian Bianchi identities.

Contribution

It provides new conditions for metric connections with skew torsion to satisfy classical curvature identities and characterizes generalized Ricci solitons with closed torsion.

Findings

Scalar curvature determined by torsion norm under certain conditions

Compact generalized gradient Ricci solitons with closed torsion are Ricci flat if torsion or scalar curvature are constant

Connections satisfying specific curvature identities are shown to be flat

Abstract

Curvature properties of a metric connection with totally skew-symmetric torsion are investigated. It is shown that if either the 3-form $T$ is harmonic, $dT=\delta T=0$ or the curvature of the torsion connection $R\in S^2\Lambda^2$ then the scalar curvature of a $\nabla$-Einstein manifold is determined by the norm of the torsion up to a constant. It is proved that a compact generalized gradient Ricci soliton with closed torsion is Ricci flat if and only if either the norm of the torsion or the Riemannian scalar curvature are constants. In this case the torsion 3-form is harmonic and the gradient function has to be constant. Necessary and sufficient conditions a metric connection with skew torsion to satisfy the Riemannian first Bianchi identity as well as the contracted Riemannian second Binachi identity are presented. It is shown that if the torsion connection satisfies the Riemannian first Bianchi identity then it satisfies the contracted Riemannian second Bianchi identity. It is also proved that a metric connection with skew torsion satisfying the curvature identity $R(X,Y,Z,V)=R(Z,Y,X,V)$ must be flat.