[Kobayashi 1972]
Definition
For a manifold $M$ the frame bundle $FM$ is a $GL(n)$-principal bundle. Given a subgroup $G$ of $GL(n)$ we call a $G$-structure on $M$ to a $G$-principal bundle $P$ which is a subbundle of $FM$.$\blacksquare$
This is a particular case of reduction of group structure with respect to the inclusion $G\to GL(n)$.
Indeed, according to Sternberg, Lectures on differential geometry:
Definition
A $G$-structure on an $n$-dimensional manifold $M$ is defined as a reduction of $FM$ to the group $G$. $\blacksquare$
When $M$ and $G$ are given, and $G$ is a closed subgroup of $GL(n)$ the existence problem becomes the problem of finding a cross section in a certain bundle. Since $GL(n)$ acts on $FM$ on the right, $G$ also acts on $FM$. If $G$ is closed the quotient $FM/G$ is also a bundle, in particular is the associated bundle to $FM$ with fibre $GL(n)/G$.
In @KobayashiNomizu1996 can be found:
Proposition 5.6. The structure group $G$ of $P(M, G)$ is reducible to a closed subgroup $H$ if and only if the associated bundle $E(M, G/H, G, P)$ admits a cross-section $\alpha: M \rightarrow E = P/H$.$\blacksquare$
Remark The correspondence is 1:1.$\blacksquare$
See the example below.
If we have a tensor field defined in a manifold $M$ it let us to define a subgroup $G$ of $GL(n)$ (the subgroup that leaves invariant the copy of the tensor in $\mathbb{R}^n$) and a $G$-structure on $M$.
I suppose the converse is true: a $G$-structure let us define one or several tensor fields in $M$. The group $G$ is possibly (I'm not sure) characterized by some tensors in $\mathbb{R}^n$, $T_i$, of type $(r_i,s_i)$, which are their _invariants_. We can cover $M$ with trivializing neighbourhoods of $E$, $\{U_{\alpha}\}$, and select the G-frames corresponding to $(x,e)\in U_{\alpha}\times G$. Finally, we can define the tensor fields whose expression in this frames is given by $T_i$.
This is better understood in the example below.
If $G=O(n)$, a $G$-structure on $M$ will be a principal bundle $E$ such that for $p\in M$ the fibre $E_p$ is made of frames such that one can be obtained from other by means of an orthogonal transformation. The choice of the G-structure implies we are pointing out in every $p$ a special kind of basis: the orthonormal ones.
On the other hand, $FM/G$ consists of families of frames that can be transformed into others in the family by means of an orthogonal transformation (orbits of the action). So it is intuitively clear that a cross section $\sigma: M\to FM/G$ (a choice of a family in every $p$) provides us with the same data that $E$ (see orthonormal frame bundle).
At the same time, this is the same as providing the metric tensor. Once we know which frames are considered orthonormal, we can cover the manifold $M$ with trivializing open sets for $E$, $\{U_{\alpha}\}$. Then define the metric $g$ like the one whose local expression in every $U_{\alpha}$ is $g|_{U_{\alpha}}\equiv \delta_i^j$.
This is not valid for a general reduction of a principal bundle because we don't have solder form??
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Author of the notes: Antonio J. Pan-Collantes
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