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Let $(M,h)$ be an hermitian manifold. I consider the Riemann curvature $R$.

$$r(X,Y,Z,T)=g(R(X,Y)Z,T)$$

$$r_{\omega}(X,Y)=r(e_i,Je_i,X,Y)$$

$J$ is the pseudo-complex structure.

Then the Hermite-Einstein equations are:

$$r_{\omega}= \lambda \omega$$

with $\omega$, the 2-form of the hermitian metric, $\lambda$ is a scalar.

Can we have spherical solutions of the Hermite-Einstein equations?

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