Metric on a Riemannian manifold $(M,g)$ is *Einstein*, if for some function $\lambda\colon M\to \mathbb R$
$$
Ric(g)=\lambda g.
$$
It is well know, that such $\lambda$ is, in fact, a constant.


The notion of Einstein metric fits perfectly into the world of K\"ahler manifolds, in this case such a metric is called *K\"ahler-Einstein*. Existence of K\"ahler-Einstein metrics is related to many deep results in differential and algebraic geometry.


I wonder whether there is something interesting going on for an arbitrary complex Hermitian manifold $(M, g, J)$. More specifically, let $\nabla$ be the Chern connection on a Hermitian manifold $(M, g, J)$ and let $\Omega\in \Lambda^{1,1}T^\*M\otimes\Lambda^{1,1}T^\*M$ be the curvature form of this connection
$$
\Omega(\xi,\bar\eta,\zeta,\bar\nu):=g\Bigl(
(
\nabla\_\xi\nabla\_{\bar\eta}-\nabla\_{\bar\eta}\nabla\_\xi-
\nabla\_{[\xi,\bar\eta]}
)\zeta, \bar\nu
\Bigr).
$$
Note that unlike the K\"ahler situation, $\nabla$ has torsion and $\Omega$ has less symmetries, then usually. In particular, we can define **two** different *Chern-Ricci forms*:
$$
\Theta^{(1)}\_{i\bar j}=g^{k\bar l}\Omega\_{i\bar jk\bar l},\quad
\Theta^{(2)}\_{k\bar l}=g^{i\bar j}\Omega\_{i\bar jk\bar l}.
$$


One might play the same game as in Riemannian case, and introduce two versions of Chern-Einstein metrics (is there a common name for these phenomena?)
$$
g\_{i\bar j}=\lambda\_1 \Theta^{(1)}\_{i\bar j}, \mbox{ and } g\_{i\bar j}=\lambda\_2 \Theta^{(2)}\_{i\bar j}.
$$


*Remark.* In the second case the metric $g$ on $T^{1,0}M$ is Hermitian-Einstein. Hermitian-Einstein is **not** necessarily Chern-Einstein for $\Theta^{(2)}$.


Very vaguely, I would like to know


**Q1** What is know about the existence of Chern-Einstein metrics on Hermitian manifolds (I am mostly interested in dimensions >2)?


**Q2** What are topological/geometrical obstructions to the existence of such metrics?


A more direct question is


**Q3** Is it true that $\lambda\_1$ and $\lambda\_2$ (whic are apriori functions on $M$) are necessarily constant?



Chern-Einstein metrics on complex Hermitian manifolds
Let me write your equations instead as
$$\Theta^{(1)}\_{i\bar{j}}=\lambda\_1 g\_{i\bar{j}},$$
$$\Theta^{(2)}\_{i\bar{j}}=\lambda\_2 g\_{i\bar{j}},$$
where $\lambda\_1,\lambda\_2$ are real-valued functions. The Chern-Ricci form $\Theta^{(1)}=\sqrt{-1}\Theta^{(1)}\_{i\bar{j}}dz^i\wedge d\bar{z}^j$ is closed.


The case $\lambda\_1\equiv 0$ is easy to deal with. Such a Hermitian metric exists if and only if the Chern-Ricci form defines the zero class in the Bott-Chern cohomology $H^{1,1}\_{\rm BC}(X,\mathbb{R})$, i.e. if and only if for some (and hence any) Hermitian metric $g$ we have $\Theta^{(1)}\_g=\sqrt{-1}\partial\bar{\partial}F$ for some smooth function $F$ on $X$. In this case the conformal metric $e^{F/n}g$ has vanishing Chern-Ricci form. The vanishing of this cohomology class implies the vanishing of $c\_1(X)$ in $H^2(X,\mathbb{R})$, but not conversely.


When $\lambda\_1$ is not identically zero, then $\lambda\_1$ is constant if and only if the torsion $1$-form $\tau$ of $g$ vanishes identically, where 
$\tau=T\_{ij}^j dz^i,$ and $T\_{ij}^k$ is the torsion of $g$. This is a result of [Goldberg.](http://www.jstor.org/stable/1969990) In this case, we must have that $g$ is in fact Kähler, since $d\Theta^{(1)}=0$.


In general, the equation $\Theta^{(2)}\_{i\bar{j}}=\lambda\_2 g\_{i\bar{j}}$ does not imply that $\lambda\_2$ is constant. Also, even if $\lambda\_2$ is constant and nonzero this does not imply that $g$ is Kähler. A simple example is the standard Hermitian metric
$$g\_{i\bar{j}}=\frac{\delta\_{ij}}{|z|^2+|w|^2},$$
on the Hopf surface $\mathbb{C}^2/((z,w)\sim (2z,2w))$, which satisfies $\Theta^{(2)}\_{i\bar{j}}=g\_{i\bar{j}}$.


In general since such metrics are Hermitian-Einstein, you get topological obstructions from the Chern number inequality.


Lastly, you could also define a third Ricci curvature
$$\Theta^{(3)}\_{i\bar{j}}=g^{k\bar{\ell}}\Omega\_{i\bar{\ell}k\bar{j}}.$$
Then the result of Goldberg also holds in this case.


