Nowhere vanishing section implies reduction of structure group

Question

Description

I noticed a repeating theme in vector bundle theory, and wonder if there's a theorem that describes this kind of phenomenon.

Given a vector bundle $E$ over a manifold $X$. If there is a nowhere-vanishing section for some associated bundle, then the structure group for $E$ can be reduced.

Example

1. Let $M^m$ be an Riemannian manifold. If there is a nowhere-vanishing section for $\Lambda^{m} {T^*M}$, then the structure group for $TM$ can be reduced from $O(m)$ to $SO(m)$.

2. Let $M^7$ be a compact oriented Riemannian manifold. If there is a nowhere-vanishing section for $\Lambda^3_+T^*M$, then the structure group of $M$ can be reduced to $G_2$. (Implicitly mentioned by a informal/use-at-your-risk note section 2.4 proposition 2.)

Question

Is there a more general theorem that depicts more examples like these?

Update (2019-09-17)

I think I have a more specific statement in mind, but lack of a rigorous proof. Given that the statement is correct, I feel that the proof can be found in some elementary textbooks about principle G-bundles. Thus, I apologize posting this question here rather than math-stack-exchange: I posted this here based on my ignorance about principle $G$-bundles.

The statement (still need a proof!) goes as follows:

Let $E\to M$ be a $G$-bundle with vector space fiber $F$, $\Phi$ a functor from ($G$-rep) to ($G$-rep), and $H$ the stabilizer group in $G$ for some fixed element $x\in\Phi(F)$. Then the followings are equivalent.

1. The structure group of $E$ can be reduced to $H$.
2. There exists a nowhere vanishing global section for $\Phi(F)\to M$.

For example, let $v$ be a nonzero vector of $\mathbb{R}^n$, its stabilzer in $GL(n)$ is isomorphic to $GL(n-1)$, so the existence of a nonvanishing section reduces the structure group from $GL(n)$ to $GL(n-1)$. Similarly, the existence of an independent $k$-frame reduces the structure group to $GL(n-k)$. If you find more comfortable with vector bundles (rather than $G$ bundles as I did), this is simply saying that there is a rank $k$ trivial subbundle in our vector bundle. Also notice that if $v$ was taken to be a zero vector, then its stabilizer is the whole group and the statement says the structure group can be reduced to itself (a null statement!).

I find this statement useful (if proved!) for it can be applied to more complicated cases when the functor $\Phi$ is not taken as an identity. For more example,

1. If $\Phi=Sym^{2,*}$, $x$ is a nondegenerated metric, then the stabilizer is the orthogonal group. Having a nonvanishing metric thus means that the structure group can be reduced to the orthogonal group.

2. If $\Phi=\Lambda^{top}$, similar arguments show that the existence of a nonvanishing top form is equivalent to "orientableness".

3. For $\Phi=\Lambda^{3,*}$, the second example in the original post shows a $G_2$ structure on a $7$-manifold.

4. In this paper, proposition $3.2$ proves that for a 7-dimensional compact manifold $M^7$, then $TM$ admitting a $Spin_7$ structure implies a $SU(2)$ structure! And the argument is to first use representation theory to create three independent spinors, whose stabilizer is $SU(2)$ at the fibre level, and to successfully reduce the structure group from the large $Spin_7$ to $SU(2)$.

Answer 1

Informally, say a type of structure $S$ which can form "bundles" over a manifold is a "symmetry structure" if all fibers are isomorphic (in an appropriate, say categorical, sense). Let $G(S)$ be the automorphism group of a single fiber (usually a Lie group). It is not hard to see that in that case an "$S$-bundle" over a manifold $X$ is given by precisely the same data as a principal $G(S)$ bundle, a.k.a. a $G(S)$ torsor.

In particular, this implies that the data of an $S$-bundle is the same as the data of an $S'$-bundle provided they have the same symmetry group. Now structures like a Riemann structure are not just bundle structures: they are bundle structures which "enrich" the tangent bundle. The tangent bundle is "a priori" a $GL_n$-structure, and other important structures are "reductions of structure" from $GL_n$ to the automorphism group of a fiber of whatever structure you are considering.

In particular, the pair $(V, g)$ for $V$ an $n$-dimensional vector space and $g\in S^2V^*$ a positive-definite bilinear form has automorhpism group isomorphic to $O(n).$ Thus choosing a positive-definite sectino of $S^2T^*_x$ consistently at all points $x$ produces a "reduction of structure" on the tangent bundle from a bundle of $GL_n$-symmetric objects to a bundle of $O(n)$-symmetric objects.

Similarly, if you fix, in addition to $g\in S^2T^*X$ a nonzero length section of $\Lambda^n(V)$ which has length $1$ w.r.t. $g$, the symmetry group decreases to $SO(n)$ (note that any nonzero section can be positively rescaled to a length-one section, hence the observation you wrote). If you fix "too much" data on top of the metric, eventually your bundle ceases to be a "symmetry structure" and you might get infinitely many non-isomorphic possibilities for your fibers. However so long as this does not happen, you will always have a group behind the scenes describing the bundle behavior.