# The Einstein-Kaehler equations

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Let $(M,g,J)$ be a Kaehler manifold ($\nabla J=J \nabla$), let $R(X,Y)$ be the riemannian curvature. I define:

$Ricc(J)=\sum_i R(J e_i, e_i)$

for an orthonormal basis $(e_i)$

$R(J) = J Ricc (J)$

$r(J)=tr (R(J))$

then I can define the Einstein-Kaehler equations:

$R(J)_{ij} - (1/2) r(J) g_{ij} =T_{ij}$

Can I reformulate the gravitation by means of these equations?

$2R(J)=JRicc(J)+Ricc(J)J$