I'm studying holomorphic vector bundles $(E,h)$ on Kähler manifolds that admit a Hermite-Einstein metrics. Particularly, I'm trying to find the motivation for the definition.
An hermitian structure $h$ on a holomorphic vector bundle $E$ is called Hermite-Einstein if $$i\Lambda_\omega F_\nabla=\lambda\cdot \text{id}_E$$ for some constant scalar $\lambda \in \Bbb R$. Here $\Lambda_\omega F_\nabla$ is the contraction by $\omega$.
What I'm currently thinking is that this is analogous to the definition of an Einstein manifold. We say that a Riemannian manifold $(M,g)$ is Einstein if $\text{Ric} = \lambda g$, i.e. the Ricci tensor is proportional to the Riemannian metric.
To this end I would like to think about $i\Lambda_\omega F_\nabla$ as somehow capturing the same information as the Ricci tensor, but I have been stuck with understanding this expression for a multiple days now.
I found an answer here stating that $\Lambda_\omega \alpha = g(\omega,\alpha)$, whenever $\alpha$ is a $2$-form or a $(1,1)$-form. Since $F_\nabla$ is a $(1,1)$-form this should give $\Lambda_\omega F_\nabla = g(\omega,F_\nabla)$, but now this deviates from what the definition stated that $\Lambda_\omega F_\nabla$ would have been a contraction by $\omega$?
If anyone here can clarify this or has some conceptual understanding about the definition, I would gladly appreciate that! Apologies if this is a question suited more for stackexchange, I did not find much about this there.
