Let $X$ be a complex manifold, and let $(E,h_E)$ and $(F,h_F)$ be Hermitian holomorphic vector bundles on $X$. Equip the holomorphic [tensor product](/page/Tensor%20Product) bundle $E\otimes F$ with the tensor product Hermitian metric $h_E\otimes h_F$. Let $\nabla^E$, $\nabla^F$, and $\nabla^{E\otimes F}$ be the Chern connections of $(E,h_E)$, $(F,h_F)$, and $(E\otimes F,h_E\otimes h_F)$, respectively, and define the curvature forms by $\Theta(E,h_E)=(\nabla^E)^2$, $\Theta(F,h_F)=(\nabla^F)^2$, and $\Theta(E\otimes F,h_E\otimes h_F)=(\nabla^{E\otimes F})^2$. Then, as an element of $\Omega^2(X,\operatorname{End}(E\otimes F))$, under the natural action of $\operatorname{End}(E)$ and $\operatorname{End}(F)$ on the tensor factors,

\begin{align*} \Theta(E\otimes F,h_E\otimes h_F)=\Theta(E,h_E)\otimes \operatorname{id}_F+\operatorname{id}_E\otimes \Theta(F,h_F). \end{align*}