  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.
