# 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 $P$.

## Facts

### Viewing a connection on a vector bundle as a principal connection

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### Transport using principal connections

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