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

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