Torsion-free $G$-Structures

Question

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

Answer 1

The bundle $P$ is made out of frames, being a subbundle of the frame bundle $F$. So each point in $P$ is a basis of a tangent space of $M$. We can take any metric on $M$, and use it to parallel transport the vectors of that basis. In general, for an arbitrary metric, such a parallel transport will take these vectors into a basis that does not belong to $P$. We don't want to write out something like $\nabla P$. We just think geometrically about carrying tangent vectors to $M$ along curves in $M$ by parallel transport. In order to see whether a given $G$-structure $P$ is torsion-free, we can pick a local section of $P$, which we think of as some 1-forms $\omega_i$. We differentiate them, to produce the connection 1-forms $\omega_{ij}=-\omega_{ji}$, so that $d \omega_i = - \omega_{ij} \wedge \omega_j$. These 1-forms $\omega_{ij}$ will split into a part valued in the Lie algebra of $G$ and a part valued in the orthogonal complement to that Lie algebra inside $SO(n)$. That orthogonal-valued bit is the torsion of the $G$-structure: if it vanishes, then parallel transport of these $\omega_i$ will act as an element of $G$ on the $\omega_i$. Therefore $P$ is taken to itself under parallel transport.

For more explanation, try Dominic Joyce's book Riemannian holonomy groups and calibrated geometry or Andreas Cap and Jan Slovak Parabolic geometries I: background and general theory.