TITLE: A question on Ricci curvature and Ricci form. 
QUESTION [7 upvotes]: It seems to me that there are two definition of Ricci flatness; real and complex ones. The real Ricci flatness claims vanishing of Ricci curvature. To define complex Ricci flatness, one needs to introduce so called the Ricci form for a Kahler manifold $(X,g)$ defined by 
$$
Ric(\omega)=-i\partial \overline{\partial}\log\det(g_{i,\overline{j}}),
$$
where $g=(g_{i,\overline{j}})$ is a Kahler metric. A Kahler manifold$(X,g)$ is called Ricci flat if $Ric(\omega)$ vanishes. 
I wonder if the complex one coincides with real one for a Kahler manifold $(X,g)$. How Ricci curvature and Ricci form are related?

REPLY [3 votes]: If you only have an almost complex manifold with a compatible metric i.e. an almost Hermitian manifold $(M,g,J)$ then you can define the Ricci form globally by $$\rho(X,Y):=Ric(JX,Y).$$ And vanishing of $Ric$, $\rho$ are equivalent. You don't need your manifold to be complex or Kähler.