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.



What are the important geometric-topological consequences of 4-dimensional version of Gauss-Bonnet-Chern theorem?
One can find some applications in Arthur Besse's book "Einstein manifolds", $\S6\ D$.


**Proposition.** If a closed manifold $M^4$ admits an Einstein metric, then $\chi(M)\ge 0$. Moreover $\chi(M)=0$ iff $M$ admits a flat metric.


*Proof.* Since $Rc(g)=\lambda g$, we have $|Rc|^2=4\lambda^2=r^2/4$. This implies $\chi(M)=1/32\pi^2 \int|Rm|^2dV\ge 0$ with equality iff $Rm=0$. $\square$


Combining with another formula for the signature of a 4-manifold
$$
\sigma(M)=\frac{1}{12\pi^2}\int\_M(|W\_+|^2-|W\_-|^2)dV,
$$
where $W\_{\pm}$ are compontens of the Weyl tensor, one can prove a stronger inequality: for a closed Einstein manifold $M^4$
$$
\chi(M)\ge 3/2|\sigma(M)|.
$$


The equality case was studied by Nigel Hitchin, [Compact four-dimensional Einstein manifolds](https://projecteuclid.org/euclid.jdg/1214432419), (1974). He proved that if $(M^4, g)$ is an Einstein manifold with $\chi(M)= 3/2|\sigma(M)|$, then $M$ is Ricci flat and either universal cover of $M$ is K3-surface, or $M$ is flat.


These are important results, since topological obstructions for the existence of Einstein metrics are very scarce. In particular in dimensions $\ge 5$ there are no known such obstructions.


