# Connection on a principal bundle

## Contents

## Definition

### Setup

Let be a differential manifold, a Lie group acting on , and a principal -bundle.

### Definition part

A **principal -connection** on this principal -bundle is a differential 1-form on with values in the Lie algebra of which is -equivariant and reproduces the Lie algebra generators of the fundamental vector fields on .

In other words, it is an element of such that:

- where denotes right multiplication by . This condition is -equivariance
- If and is the fundamental vector field corresponding to , then identically on .

## 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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