Intuition of Chern-Weil theory

Let $ P \rightarrow M$ be a $G$-principal bundle. The lie algebra of $G$ is $\frak{g}$ and $P$ has connection form $\omega \in H^1(P,\frak{g})$ and curvature form $\Omega \in H^2(P,\frak{g})$. We consider $I^*(G)$ the set of invariant polynomials of $G$ ie the multilinear functions $f: \frak{g} \times ... \times \frak{g} \rightarrow \mathbb{R}$ satsifying

$f(Ad(g)X_1,...,Ad(g)X_k) = f(X_1,...,X_n)$

for all $g \in G$ with $Ad(g) = (R_g)_*$ the map induced on $\frak{g}$ by the right multiplication with an element of $G$. It turns out very surprisingly that

1)the form

$f(\Omega)(V_1,...,V_{2k}) = f(\Omega(V_1,V_2),...,\Omega(V_{2k-1},V_{2k}))$

can be projected to a form in $H^{2k}(M)$ is independant of the choice of connection for $P$ and

2) in the case of the $GL(n,\mathbb{K})$-bundle (with $\mathbb{K} = \mathbb{R} $ or $ \mathbb{C}$) associated to a vector bundle this form represents a characteristic class of the bundle.

My problem is that I have read proofs of the statements but I do not feel I understand why they are true. In other words, what made Chern and Weil expect that this construction would yeild characteristic classes?

Solution 1:

Chern-Weil theory is a vast generalization of the classical Gauss-Bonnet theorem.

The Gauss-Bonnet theorem says that if $\Sigma$ is a closed Riemannian $2$-manifold with Gaussian curvature $K$, then $$\int_\Sigma K \,dA = 2\pi \chi(\Sigma).$$ In the 1940's Chern generalized this result to all even-dimensional Riemannian manifolds $M$: if $\dim(M) = 2n$ and $\Omega$ is the curvature form of the Levi-Civita connection on $M$, then $$\int_M \mathrm{Pf}(\Omega) = (2\pi)^n \chi(M).$$

After proving the above Chern-Gauss-Bonnet formula, Chern had the insight that the formula is about a particular $\mathrm{SO}(2n)$-bundle (the frame bundle of the tangent bundle of $M$) and a particular connection on that bundle (the Levi-Civita connection). At this time, I believe some other general facts from Chern-Weil theory were understood in the case of the tangent bundle with the Levi-Civita connection, for example that $H^\ast(B\mathrm{SO}(2n); \Bbb R) \cong I^\ast(\mathrm{SO}(2n))$.

Chern's next step was to shift the point of view towards fixing $M$ and varying the principal $G$-bundle and choice of connection. From this idea, Chern-Weil theory was developed.