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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?

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    $\begingroup$ Good question! Basically, Chern–Weil formulas come from (Lie algebra) cohomology of $g$ and its close relative $(S^\ast g[2])^{g}$ as an algebraic model for the universal bundle $EG\to BG$. (I hope to write an answer later.) $\endgroup$
    – Grigory M
    Dec 25, 2013 at 17:53

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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.

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    $\begingroup$ thank you this is very interesting but dou you have any insight as to why the evaluation of an invariant polynomial on the curvature form would be independant of the choice of connection? $\endgroup$
    – Vincent L.
    Oct 14, 2013 at 2:13
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    $\begingroup$ I think the invariant form has to be independent of the local framing. The evaluation of the invariant polynomial would be dependent on the connection, since different connection give you different curvature. $\endgroup$ Aug 3, 2014 at 23:32
  • $\begingroup$ Different connections do give different curvatures, but that only changes the invariant polynomial by an exact piece. This does not change the cohomology class of the invariant polynomial and hence does not change its periods. $\endgroup$ Oct 9, 2015 at 12:54

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