---
tag: VYNQVZ8XVF
kind: concept
name: G-structure
msc: [53C10]
specializes: [8E37E74XVP]
sources:
  - url: https://en.wikipedia.org/wiki/G-structure_on_a_manifold
same_as:
  wikidata: Q5511959
  wikipedia: G-structure on a manifold
---

## Summary

A $G$-structure on an $n$-manifold is a choice, at every point, of a preferred family of frames of the tangent space.
Any two preferred frames at a point are related by an element of a fixed subgroup $G$ of $\mathrm{GL}(n, \R)$.
Formally it is a reduction of the [[frame bundle|F4HWTRNXAP]] to the group $G$.
Many geometric structures are $G$-structures: Riemannian metrics for $G = \mathrm{O}(n)$, orientations for $\mathrm{GL}^+(n, \R)$, almost complex structures for $\mathrm{GL}(m, \C)$ with $n = 2m$.
The notion gives one language for comparing and classifying these structures.

## Explanations

### L1 human

At each point of a manifold one can choose many bases of the tangent space.
A geometric structure picks out some of them as the good ones.
A Riemannian metric picks out the orthonormal bases, and any two of these differ by a rotation or reflection, an element of $\mathrm{O}(n)$.
An orientation picks out the positively oriented bases, which differ by matrices of positive determinant.
A $G$-structure is this idea for an arbitrary group $G$.