---
tag: F4HWTRNXAP
kind: concept
name: Frame bundle
msc: [58A32, 53C10, 55R10]
specializes: [8E37E74XVP]
sources:
  - url: https://en.wikipedia.org/wiki/Frame_bundle
same_as:
  wikidata: Q5477902
  wikipedia: Frame bundle
---

## Summary

The frame bundle of a manifold is the space of all ordered bases of all tangent spaces.
A point of it is a point of the manifold together with a choice of coordinate axes for the tangent space there.
The general linear group acts on it by changing the basis, which makes it a principal bundle.
It is the natural home of connections and of G-structures, such as Riemannian metrics and orientations.

## Explanations

### L1 human

At each point of a surface you can draw two arrows tangent to it that are not parallel; they form a frame.
The frame bundle collects all possible frames at all points.
Two frames at the same point differ by an invertible $2 \times 2$ matrix.
If you only allow perpendicular arrows of unit length, you get the smaller orthonormal frame bundle.