---
tag: 8E37E74XVP
kind: concept
name: Principal bundle
msc: [55R10, 53C05]
specializes: [CX4M8K61AB]
sources:
  - url: https://en.wikipedia.org/wiki/Principal_bundle
same_as:
  wikidata: Q1365548
  wikipedia: Principal bundle
---

## Summary

A principal bundle is a fiber bundle whose fibers are copies of a group $G$.
The group $G$ acts on the total space and moves points within each fiber.
Each fiber is like the group itself, but without a chosen identity element.
Frame bundles of manifolds and the Hopf fibration are standard examples.
Connections on principal bundles are the language of gauge theory, and every vector bundle can be built from a principal bundle.

## Explanations

### L1 human

Over each point of a manifold, consider all possible coordinate frames for the tangent space.
No frame is preferred, but any two differ by exactly one invertible matrix.
So the frames over one point look like the group of invertible matrices, with no special element.
Collecting these sets over all points gives a principal bundle, the frame bundle.