TITLE: What are the important geometric-topological consequences of 4-dimensional version of Gauss-Bonnet-Chern theorem?
QUESTION [13 upvotes]: 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.

REPLY [4 votes]: A good paper in this direction is "Some implications of the generalized Gauss-Bonnet theorem" written by Bishop  and Goldberg  which proved the following  two theorems

Theorem 1.1. A compact and oriented Riemannian manifold of dimension $4$
  whose sectional curvatures are non-negative or nonpositive has non-negative
  Euler-Poincare characteristic. If the sectional curvatures are always positive
  or always negative, the Euler-Poincare characteristic is positive.
Theorem 1.2. In order that a $4$-dimensional compact and orientable manifold
  M carry an Einstein metric, i.e., a Riemannian metric of constant Ricci
  or mean curvature $R$, it is necessary that its Euler-Poincare characteristic be
  non-negative.

and a corollary

Corollary 1.2. If $V$ is the volume of $M$,
  $$\chi(M)\geq\frac{VR^2}{12\pi^2}$$
  equality holding if and only if $M$ has constant curvature.