I take an hermitian manifold $(M,g,J)$ and I define from the riemannian curvature $R(X,Y)$ as a $2$-form: $$ Ricc(J)= \sum_i R(J e_i,e_i) $$ with $(e_i)$ an orthonormal basis of the tangent. $$ 2R(J)=J Ricc(J)+ Ricc(J)J $$ $$ r(J)=tr(R(J)) $$ The hermitian Einstein metric verifies: $$ R(J)_{ij}- \frac{1}{2} r(J)g_{ij}=0 $$ Have you references about such a metric?


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