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.

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 constructing Characteristic classes.

Is Weil homomorphism introduced (and used) only to construct 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.

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?

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.