Let $G$ be a Lie group and $\pi:P\rightarrow M$ be a principal $G$ bundle.

The notion of reduction of structure group is standard but I will recall here in case some one needs it.

Let $f:P(M,G)\rightarrow P'(M',G')$ be a morphism of principal bundles such that $f:P\rightarrow P'$ is an imbedding and $f:G\rightarrow G'$ is a monomorphism. If $M=M'$ abd the induced map $f:M\rightarrow M'$ is identity map, we call $P(M,G)$ to be reduced bundle for $P'(M,G')$ .

Given a principal bundle $P’(M’,G’)$ and a Lie subgroup $G$ of $G’$, we say the structure group $G’$ is reduced to $G$ if there is a reduced bundle $P(M,G)$.

Reduction of structure group says some thing interesting about manifolds involving it. For example,

- A manifold
**admits an almost-complex structure**if the frame bundle on the manifold, whose fibers are $GL(2n,\mathbb{R})$, can be reduced to the group $GL(n,\mathbb{C})\subset GL(2n,\mathbb{R})$. - A manifold is
**orientable**if and only if its frame bundle can be reduced to the special orthogonal group, $SO(n,\mathbb{R})\subset GL(n,\mathbb{R})$.

I am interested in knowing similar results about reduction of structure group. Please add references (if possible, a sketch of the proof) for results you quote here. One result in one answer please.

anygroup homomorphism $\phi\colon H\to G$ and a principal $G$-bundle $P\to M$ (or even a $G$-fibre bundle), one can ask for the existence of a principal $H$-bundle $Q\to M$ such that $P \simeq Q\times_H G$, where $H$ acts on $G$ via $\phi$. This is equivalent to asking if the Čech cocycle arising from the transition functions for $P$ is in the image of $H^1(M,H) \to H^1(M,G)$. There are a large number of other equivalent formalisations of this. For the fibre bundle case, the version using transition functions is the one that makes immediate sense. $\endgroup$ – David Roberts Aug 26 '18 at 23:56