Geometry of $G$-Structures via the Intrinsic Torsion

We study the geometry of a $G$-structure $P$ inside the oriented orthonormal frame bundle $\mathrm{SO}(M)$ over an oriented Riemannian manifold $M$. We assume that $G$ is connected and closed, so the quotient $\mathrm{SO}(n)/G$, where $n=\dim M$, is a normal homogeneous space and we equip $\mathrm{SO}(M)$ with the natural Riemannian structure induced from the structure on $M$ and the Killing form of $\mathrm{SO}(n)$. We show, in particular, that minimality of $P$ is equivalent to harmonicity of an induced section of the homogeneous bundle $\mathrm{SO}(M)\times_{\mathrm{SO}(n)}\mathrm{SO}(n)/G$, with a Riemannian metric on $M$ obtained as the pull-back with respect to this section of the Riemannian metric on the considered associated bundle, and to the minimality of the image of this section. We apply obtained results to the case of almost product structures, i.e., structures induced by plane fields.