# Chern form

From Diffgeom

## Definition

### Setup

Let be a differential manifold, a complex vector bundle over , and a connection on . Let denote the Riemann curvature tensor of , and denote the rank of .

### Definition part

The **Chern form** of the pair , denoted , is defined as:

is even-dimensional, viz it is a sum of elements from even-dimensional cocycles. The member from the cochain group is denoted . Special formulae exist for the first and last member:

and:

The Chern-Weil theorem gives the basic facts about the Chern form that also establish its importance. In particular, it shows that the cohomology classes of the Chern forms are independent of the choice of connection.