Csdrhrt

Let $M$ a manifold and $G$ a group.

Is it true that the statement: "the structure group of the frame bundle of $M$ can be reduced to $G$" simply means:

There is a subbundle $S$ of the frame bundle (i.e. at each point some distinguished set of frames, e.g. the orthogonal ones) and an atlas of $M$ such that the transition functions map $S$ to $S$?

If yes, why is this equivalent to $M/G$ having a global section?

Let $M$ be an $n$-dimensional manifold and $E_M$ it bundle of frames. Recall that if $x$ is an element of $M$, and $T_xM$ its tangent space, a linear frame at $x$ is an inveritble linear map $mathbb{R}^nrightarrow T_x$. We denote by $E_M(x)$ the set of linear frames at $x$, and by $E_Mcup_{xin M}E_M(x)$, $E_M$ is a $Gl(n,mathbb{R})$-principal bundle. It can be reduced to $Gsubset Gl(n,mathbb{R}$ is equivalent to saying that there exists a trivialisation $(U_i,g_{ij})$ of $E_M$ such that $g_{ij}:U_icap U_jrightarrow G$, this implies the existence of a $G$-principal bundle $E_Grightarrow M$ whose coordinate change are $(U_i,g_{ij})$.

Consider $E_M/G$, it is the bundle over $M$ which is obtained by gluing $U_itimes Gl(n,mathbb{R})/G$ with $U_icap U_jtimes Gl(n,mathbb{R})/G rightarrow U_icap U_jtimes Gl(n,mathbb{R})/G$ $(x,y)rightarrow (x,bar g_{ij}(x)(y))$ where $bar g_{ij}(x)$ is the map induced by $g_{ij}(x)$ on $Gl(n,mathbb{R})/G$ by $g_{ij}(x)$. If there exists a $G$-reduction, $g_{ij}(x)in G$ and $bar g_{ij}(x)$ is the identity, we deduce that $E_M/G$ is trivial, and henceforth has a global section.