Differential Forms

Let $M$ be a manifold. There are bunch of definitions, but I will use the following definition.

Definition

The differential form on $M$ is a smooth section of the cotangent bundle over $M$. We denote it as ${\Omega }^{\ast }(M)$ or $\Gamma ({\bigwedge }^{\ast }{T}^{\ast }M)$.

Exterior Product

${\Omega }^{\ast }(M)$ is an algebra with exterior product.

Exterior Derivative

Application

Lie algebra valued 1-form on a principal bundle is a connection on the manifold $M$.