The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important application on Riemann surface such as A 2-torus cannot carry a Riemannian metric with positive sectional curvature. The generalized Gauss–Bonnet theorem (Gauss-Bonnet-Chern) in dimension $n=4$, for a compact oriented manifold states that $$\chi(M)=\frac{1}{32\pi^2}\int_M\left(|Rm|^2-4|Rc|^2+r^2\right)d\mu,$$ where $Rm$ is the full Riemann curvature tensor, $Rc$ is the Ricci curvature tensor, $r$ is the scalar curvature. My question is

What are the important local-global results of 4-dimensional version of Gauss-Bonnet-Chern theorem similar to 2-Dimensional case?

Thanks.