Chern's definition of G-structure

Question

Section 4, page 10 of The geometry of G-structures by S. S. Chern, Bull. Amer. Math. Soc. 72(2): 167-219 (March 1966), the definition of G-structure is somewhat vague, and I have have the impression that it is wrong.

Let $T$ be an $n$-dimensional real vector space and let $T^*$ be its dual space. Denote their pairing by $\langle y, \xi \rangle \in R$, $y \in T$, $\xi \in T^*$. We let $GL(n,R)$ act on $T$ on the left and on $T^*$ on the right, so that the following relation holds:

$$\langle gy, \xi \rangle = \langle y, \xi g \rangle, \quad \quad g \in GL(n,R).$$

The tangent bundle over $M$ has the local charts $(x,y_U)$, $x \in U$, $y_U \in T$, which are the local coordinates of the tangent vectors relative to $U$. The local coordinates $(x, y_U)$ and $(x, y_V)$ in $U \cap V$ define the same tangent vector if and only if $y_U = g_{UV}(x)y_V$, where $g_{UV} : U \cap V \to GL(n,R)$. Consider a subgroup $G$ of $GL(n,R)$; we say that the structural gropu of the tangent bundle is reduced to $G$, if all $g_{UV}(x) \in G$. Such a reduction will simply be called a $G$-structure.

I cannot relate this definition to other ones found elsewhere (using the notion of principal G-bundles). Is it complete? Correct? What am I missing?

The article by Chern is in the reference list of Wikipedia's article G-Structures on a manifold

Answer 1

Chern's definition is a little sloppy. His phrasing here gives the impression that reduction of the structure group is a property of $M$, which it's not; reductions need not be unique if they exist. You can see that reflected implicitly here by the fact that local coordinates and hence transition functions are highly non-unique. It's also a bit confusing that even if reductions exist in some coordinates they may not be witnessed by a generic choice of coordinates.

To fix it one can say that a choice of reduction of the structure group is a choice of local coordinates such that the transition functions lie in $G$; up to isomorphism this agrees with the usual definition in terms of $G$-bundles, although the $G$-bundle definition is more general because the map $G \to GL_n(\mathbb{R})$ need not be injective (e.g. $G$ might be the spin group $\text{Spin}(n)$).

Answer 2

I validated the answer of Qiaochu Yuan as the accepted answer because it helped me recover what is implicitly meant in the quoted text.

I add this answer as a detailed explanation of the source of confusion, I think it can help other people.

1. When Chern wrote

The tangent bundle over $M$ has the local charts $(x,y_U)$, $x∈U$, $y_U∈T$, which are the local coordinates of the tangent vectors relative to $U$.

I thought he meant canonical charts of the tangent bundle, i.e. charts that are associated to a diffeomorphism $\phi: U\to U'\subset \mathbb{R}^n$ and where $(x,v)$, $x\in M$, $v\in T_x M$ is mapped to $(\phi(x),(D_x\phi)(v))\in \phi(U)\times \mathbb{R}^n$. Note that the action of $GL(n,\mathbb{R})$ introduced at the beginning yields an identification from $T$ to $\mathbb{R}^n$, and $\phi$ allows to identify $U$ and $U'$, so we can identify $U'\times \mathbb{R}^n$ with $U\times T$.

With this belief, the definition looked wrong: for instance Riemannian metrics are supposed to correspond to $G=O(2)$. Take the unit Euclidean sphere centred on $0$ in Euclidean space $\mathbb{R}^3$ with the induced Riemannian metric. With this belief, $g_{UV}$ is the composition of the differential of two manifold charts. Consider two charts on, for instance $(x,y,z)\mapsto(x,y)$ when $z>0$ and $(x,y,z)\mapsto(y,z)$ when $x>0$. The change of charts is $(x,y)\mapsto(y,\sqrt{1-x^2-y^2})$, and its differential at $(x,y)$ does not belong to $O(2)\subset GL(2,\mathbb{R})$.

Still with this belief, a way to fix it would be to ask that on every canonical chart one chooses for all $x\in U$ a frame $f_{U}(x): \mathbb{R}^n \to T$, with the compatibility condition $$ f_{V,x}^{-1} \circ g_{UV}(x) \circ f_{U,x} \in G . $$

2. However, what he actually meant is a vector bundle chart, in the sense of this link.

This may sound even more confusing before one understands that he is not allowing us to take all vector bundle charts. If he would, then the discussion above would still apply, just worse.

Actually he meant to define a $G$-structure via a special atlas on the tangent bundle seen as a general vector bundle, such that the transition maps between fibres belong to $G$; this is the spirit of $G$-bundles, see this link. This is called a $G$-atlas. This is in the spirit of the difference between topological manifold, smooth manifold, complex manifold, etc., where one asks transition maps to belong to different classes. In particular, here, two $G$-atlases are said to define the same $G$-structure whenever they are compatible, and this is equivalent to say that their union is still a $G$-atlas.

3. I will end this description by relating it to another definition of $G$-structure that I recomposed by reading various sources.

Given a manifold $M$ of dimension $n$, and a subgroup $G$ of $GL(n,R)$ (*), a $G$-structure is the choice for each $x\in M$ of a subset $F_x$ of the set $\cal F(\mathbb{R}^n,T_x M)$ of frames $f:\mathbb{R}^n \to T_x M$ ($f$ bijective linear map) with the condition that $F_x$ is a whole and single orbit under the action of $G$ on $\cal F(\mathbb{R}^n,T_x M)$ by right-multiplication (pre-composition) by $g\in G$.

Continuity/smoothness conditions can be added.

(*) Often $G$ is a Lie subgroup of $GL(n,R)$; on the other hand, one can generalize the notion where $G$ is any group and were we take a group morphism from $G$ to $GL(n,R)$

The relation is as follows: the set $F_x$ is of the form $f G = \{f \circ g; f\in \cal F(\mathbb{R}^n,T_x M),\, g\in G\}$ where $f$ is not unique (it can be replaced by any $f\circ g$). One can locally choose a particular $f$ for every $x$, in a continuous/smooth fashion and this defines a local trivialization (vector bundle chart) of the tangent bundle, and those trivializations satisfy Chern's condition.

$g_{UV}(x) \in G$

TL;DR

Chern is not using canonical charts of the tangent bundle but defining a $G$-structure as a $G$-atlas in the sense of $G$-bundles.