---
tag: GA7437914V
kind: definition
name: Connection on a principal bundle
msc: [53C05]
defines: NJ21PN5ZGJ
uses: [8E37E74XVP, 0E3G0GRPZ6, TD6VEP3SCE]
sources:
  - url: https://en.wikipedia.org/wiki/Connection_(principal_bundle)
---

## Definition

Let $G$ be a Lie group with Lie algebra $\mathfrak{g}$, and let $\pi : P \to M$ be a smooth [[principal $G$-bundle|8E37E74XVP]] with right action $R_g(u) = u \cdot g$.
For $X \in \mathfrak{g}$, the fundamental vector field $X^\#$ on $P$ is $X^\#_u = \frac{d}{dt}\big|_{t=0} \, u \cdot \exp(tX)$, where $\exp$ is the exponential map.
A *connection form* on $P$ is a smooth $\mathfrak{g}$-valued [[differential $1$-form|0E3G0GRPZ6]] $\omega$ on $P$ such that:

1. $\omega(X^\#) = X$ for every $X \in \mathfrak{g}$.
2. $R_g^* \omega = \mathrm{Ad}(g^{-1}) \, \omega$ for every $g \in G$, where $\mathrm{Ad}$ is the adjoint representation.

A *connection* on $P$ is a connection form on $P$.

## Remarks

Equivalently, a connection is an [[Ehresmann connection|TD6VEP3SCE]] $H \subset TP$ with $dR_g(H_u) = H_{u \cdot g}$ for all $u$ and $g$; the two notions correspond by $H = \ker \omega$.
Example: on the trivial bundle $P = M \times G$, let $\theta$ be the left-invariant Maurer–Cartan form of $G$, so $\theta_g(v) = dL_{g^{-1}}(v)$, and let $\omega = \mathrm{pr}_2^* \theta$.
Clause 1 holds since $X^\#_{(p, g)} = (0, dL_g X)$, and clause 2 holds since $R_h^* \theta = \mathrm{Ad}(h^{-1}) \theta$.
Non-example: $\omega = 0$ satisfies clause 2 but fails clause 1 when $\dim G > 0$.
For a matrix group the trivial example reads $\omega = g^{-1} dg$.
The difference of two connection forms is a $\mathfrak{g}$-valued $1$-form that vanishes on vertical vectors and satisfies clause 2; so connections on $P$ form an affine space.