In Kobayashi and Nomizu's book Foundations of Differential geometry they introduce the concept of connection on a principal $G$ bundle. In this book, they use connection on a principal bundle to define (among other things) the characteristic classes of a (complex) vector bundle.


Let $G$ be a Lie group and  $P(M,G)$ be a principal $G$ bundle. Let $\Gamma$ be a connection on $P(M,G)$. This defines what is called a Weil homomorphism $I(G)\rightarrow H^*(M,\mathbb{R})$.

 
Given a complex vector bundle $E\rightarrow M$ with fibre $\mathbb{C}^r$ they consider associated principal $Gl(r,\mathbb{C})$ bundle $P\rightarrow M$ and define $k$-th Chern class of $E$ to be image of some element of $I(G)$. 

But, it seems this Weil homomorphism can do more than defining Characteristic classes.

> Is Weil homomorphism introduced (and used) only to define Characteritic classes? If not, where else do we use this Weil homomorphism?

Edit (2 Nov 2019) : While searching for something, I have found what is called a **refined Chern-Weil homomorphism**, which has something to do with secondary characteristic classes.