The Hermitian metric $H$ along the fibers of the holomorphic vector bundle $E$ over a K\"ahler manifold $(M,\omega)$ is **weak Hermitian-Einstein** if $$\sqrt[]{-1}\Lambda_\omega F_H =
\phi I$$ at every point of $M$, where $\gamma$ is a function on $M$ and $I$ is the identity
endomorphism of $E$ and $F_H$ is the Chern curvature form of $(E, h)$. Here $\Lambda_\omega$ is the contraction with $\omega$ which means 
$$F_H\wedge\omega^{n-1}=(n-1)!\Lambda_\omega F_H\frac{\omega^n}{n!}$$

Now take $E=TM$, then we can define weak Kahler-Einstein metric as $Ric(\omega)=\phi \omega$ and hence $c_1(TM)=[Ric(\omega)]=[\phi \omega]$



From Kobayashi book (complex vector bundle ) we have the following theorem

**Theorem**: An Hermitian vector bundle $(E, h)$ over a Riemann surface
$(M, g)$ satisfies the weak Einstein condition if and only if it is projectively flat.


Note that in general one direction always works

**Theorem**: Let $(M, g)$ is a Kahler manifold, Let $(E, h)$ be a projectively flat vector bundle over $M$.
Then, for any Hermitian metric $g$ on $M$,$(E, h)$ satisfies the weak Einstein
condition.

We have also the following theorem.

**Theorem**: If an Hermitian vector bundle $(E, h)$ over a compact Kahler
manifold $(M, g)$ satisfies the weak Einstein condition with factor $φ$, there is a conformal change $h −→ h′ = ah$ such that $(E, h′ )$ satisfies the Einstein condition with a constant factor $c$. Such a conformal change is unique up to a
homothety. Now you can take $E=TM$