The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important applications to Riemann surface theory. 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 the 4-dimensional version of Gauss-Bonnet-Chern theorem similar to the 2-Dimensional case?

The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important applications to Riemann surface theory. 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 the 4-dimensional version of Gauss-Bonnet-Chern theorem similar to the 2-Dimensional case?

Update:(2,September,2017)

This update is just an additional information. If $W$ denote the Weyl curvature of $(M,g)$ then $${32\pi^2}\chi(M)=\int_M(|W|^2+ 8Q_g)d\mu,$$ where $$Q_g:=-\frac{1}{12}(\Delta_gr-r^2+3|Rc|^2),$$ is the Paneitz $Q$ curvature introduced by Branson.

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. 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.

The Gauss–Bonnet theorem in differential geometry is an important statement about surfaces which connects their geometry to their topology and has very important applications to Riemann surface theory. 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 the 4-dimensional version of Gauss-Bonnet-Chern theorem similar to the 2-Dimensional case?