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  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?

asked Aug 19, 2018 in Theoretical Physics by Antoine Balan (80 points) [ revision history ]
edited Aug 21, 2018 by Antoine Balan
If I am not mistaken, your Ricc(J) is the Ricci curvature 2-form (obtained from the usual Ricci curvature as the Kähler form is obtained from the metric). In particular, R(J)_ij is antisymmetric in ij, whereas g_ij and T_ij are symmetric in ij so your equation seems trivial. To obtain a reformulation of Einstein equation, you need to replace g_ij by \omega_ij (and T_ij by the associated 2-form).

Ricc(J) isn't just obtained from the usual Ricci curvature, moreover it is antisymmetric, so that R(J) is symmetric.

For an hermitian metric, but not necessary complex (almost-complex), I propose to take:

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

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