Chern-Weil theorem

Statement

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

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

$c_k(E,{\mathrm{\nabla }}^1)-c_k(E,{\mathrm{\nabla }}^0)=dT({\mathrm{\nabla }}^1,{\mathrm{\nabla }}^0)$

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