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 $P$.

Related notions

• Connection on a vector bundle
• Ehresmann connection
• Linear connection

Facts

Viewing a connection on a vector bundle as a principal connection

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Transport using principal connections

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