Let (M,h) be an hermitian manifold. I consider the Riemann curvature R.
r(X,Y,Z,T)=g(R(X,Y)Z,T)
rω(X,Y)=r(ei,Jei,X,Y)
J is the pseudo-complex structure.
Then the Hermite-Einstein equations are:
rω=λω
with ω, the 2-form of the hermitian metric, λ is a scalar.
Can we have spherical solutions of the Hermite-Einstein equations?