Chern-Weil theorem

Statement

Let $c$ denote the Chern form, viz $c(E,\nabla )$ is the Chern form for vector bundle $E$ and connection $\nabla$ on $E$. Then:

• $c_{k}(E,\nabla )$ are closed for all $k$ and all $\nabla$
• For all positive integers $k$, there exists a $(2k-1)$ form $T$ such that for all $\nabla ^{0},\nabla ^{1}$ on $E$, we have:

$c_{k}(E,\nabla ^{1})-c_{k}(E,\nabla ^{0})=dT(\nabla ^{1},\nabla ^{0})$

Thus, $c_{k}$ defines an element in $H^{2k}(M,\mathbb {R} )$ independent of $\nabla$. This elements is termed the $k^{th}$ Chern class (the word class because it is a cohomology class).