I have the following question. Let $G \subset SO(n)$ be a Lie Group and $M$ be a smooth manifold of dimension $n$. Furthermore let $P$ be a $G$-structure on $M$ i.e. $P$ is a principal subbundle of the principal $GL(n,\mathbb{R})$-bundle $F$ (the frame bundle) on $M$. Set now $S=F/G$ which carries a smooth manifold structure with the quotient mapping $\tau : F \rightarrow S$ and $\overline{\pi}:S\rightarrow M$ and $\pi:F\rightarrow M$ the obvious bundle projections. It is known that $G$-Structures on $M$ are in one to one correspondence with sections of $\overline{\pi}:S\rightarrow M$. Since $G \subset SO(n)$ it is also known that every $G$-Structure has an underlying Riemannian metric, denote it by $g_{\sigma}$, for a section $\sigma : M \rightarrow S$ and an orientation that is defined by the condition that $u:T_{x}M \rightarrow \mathbb{R}^{n}$ be an oriented isometry for all $u \in P_{x}$ and all $x \in M$. What I just wrote is only some preliminary work. Now to my real question on the understanding of the definition of a torsion-free $G$-Structure. The definition is as follows: A $G$-Structure $P$ on $M$ and a the corresponding section $\sigma : M \rightarrow S$ are said to be torsion-free if $P$ is parallel with respect to the Levi-Civita connection of the underlying Riemannian metric.

I dont understand this definition. How can a principal bundle be parallel with respect to the Levi-Civita connection of the underlying Riemanninan metric, i.e. how can one understand $\nabla P = 0$, where $\nabla$ is the Levi-Civita connection of $g_{\sigma}$ on $M$ ? How can one differentiate a principal bundle with a connection that "lives" on $M$ ? I hope someone can explain this to me.

Greetings monica