Connection on a principal bundle

Definition

Setup

Let $M$ be a differential manifold, $G$ a Lie group acting on $M$ , and $\pi: P \to M$ a principal $G$ -bundle.

Definition part

A principal $G$ -connection on this principal $G$ -bundle is a differential 1-form on $P$ with values in the Lie algebra $\mathfrak{g}$ of $G$ which is $G$ -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on $P$ .

In other words, it is an element $\omega$ of $\Omega^1(P,\mathfrak{g})$ such that:

$Ad(g)(R_g^*\omega) = \omega$ where $R_g$ denotes right multiplication by $g$ . This condition is $G$ -equivariance
If $\xi \in \mathfrak{g}$ and $X_\xi$ is the fundamental vector field corresponding to $\xi$ , then $\omega(X_\xi) = xi$ identically on $P$ .