If we consider a spin manifold $M$, we can define the Ricci curvature $Ricc (X,Y)$ which is a symmetric tensor, moreover the spinors are defined, so that we can define a Dirac-Ricci operator: $$DR(\psi)=\sum_{ij}Ricc(e_i,e_j)[ e_i. \nabla_{e_j}\psi+e_j .\nabla_{e_i}\psi]$$ with $(e_i)$ an orthonormal basis.

If the manifold $M$ is an Einstein manifold ($Ricc=\lambda g$), then $DR=\lambda D$, the Dirac-Ricci operator is the Dirac operator $D$. Have we nice properties of such an operator in the general case?