TITLE: What does reduction of structure group of principal bundle say?
QUESTION [7 upvotes]: 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 $M=M'$ and 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.

REPLY [4 votes]: Let $G$ be a topological group and $M$ be a smooth manifold. Then, a reduction of the structure group of the frame bundle from $\mathrm{GL}_n(\mathbb R)$ to $\mathrm{GL}_n(\mathbb R)\times G$ is equivalent data to a principal $G$-bundle $Q\to M$.
Here's a proof sketch: if $H$ and $H'$ are groups, an $(H\times H')$-torsor is the same thing as a product of an $H$-torsor and an $H'$-torsor. So the fiber of the principal $(\mathrm{GL}_n(\mathbb R)\times G)$-bundle we obtained from the frame bundle at some $x\in M$ is a product of a $\mathrm{GL}_n(\mathbb R)$-torsor $P_x$ and a $G$-torsor $Q_x$, and $P_x$ is the $\mathrm{GL}_n(\mathbb R)$ of bases of $T_xM$. Since $Q_x$ varies smoothly, it defines a principal $G$-bundle $Q\to M$.