Chern class via Chern–Weil theory

Let $M$ be a smooth manifold and let $\pi:E\to M$ be a complex vector bundle of rank $n$ equipped with a (linear) connection $\nabla$. Write $F_\nabla\in\Omega^2(M;\mathrm{End}(E))$ for its curvature .

Definition (Chern forms and Chern classes)

The total Chern form of $\nabla$ is the even differential form

where the determinant is computed fiberwise after identifying $\mathrm{End}(E)$ with matrices in local frames. Expanding by degree gives

Then:

1. Each $c_k(\nabla)$ is closed, i.e. $d\,c_k(\nabla)=0$, where $d$ is the exterior derivative .
2. The de Rham cohomology class $[c_k(\nabla)]\in H^{2k}_{\mathrm{dR}}(M)$ is independent of the choice of $\nabla$.
3. The $k$th Chern class $c_k(E)\in H^{2k}(M;\mathbb Z)$ is the unique integral cohomology class whose image under the natural map $H^{2k}(M;\mathbb Z)\longrightarrow H^{2k}(M;\mathbb R)\cong H^{2k}_{\mathrm{dR}}(M)$ equals $[c_k(\nabla)]$.

Equivalently, $c_k(E)$ is the characteristic class obtained by the Chern–Weil construction for the structure group $U(n)$ of a Hermitian bundle (or the corresponding principal bundle of unitary frames) using the invariant polynomial given by the $k$th elementary symmetric function of eigenvalues.

The Chern classes are natural under pullback: for any smooth map $f:N\to M$,

Examples

1. Trivial bundle. If $E\cong M\times\mathbb C^n$ admits the flat connection ($F_\nabla=0$), then $c(\nabla)=1$ and hence $c_k(E)=0$ for all $k\ge 1$.

2. Complex line bundle. If $\mathrm{rank}_{\mathbb C}E=1$, then

   $$c(\nabla)=1+\frac{i}{2\pi}F_\nabla,$$

   so $c_1(E)$ is represented in de Rham cohomology by the real 2-form $\frac{i}{2\pi}F_\nabla$.

3. Whitney sum behavior (curvature-level). If $E=E_1\oplus E_2$ with a block-diagonal connection $\nabla=\nabla_1\oplus\nabla_2$, then $F_\nabla$ is block-diagonal and

   $$c(\nabla)=c(\nabla_1)\wedge c(\nabla_2),$$

   recovering the usual multiplicativity of total Chern classes under direct sum.