Geometry of G-Structures via the Intrinsic Torsion

Abstract

We study the geometry of a G-structure P inside the oriented orthonormal frame bundle SO(M) over an oriented Riemannian manifold M. We assume that G is connected and closed, so the quotient SO(n)/G, where n=dimM, is a normal homogeneous space and we equip SO(M) with the natural Riemannian structure induced from the structure on M and the Killing form of SO(n). We show, in particular, that minimality of P is equivalent to harmonicity of an induced section of the homogeneous bundle SO(M)×SO(n)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.