Geometry of G-Structures via the Intrinsic Torsion

Abstract

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 = dim M , 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.

Examples of Minimal $$\varvec{G}$$ G -structures

“…where S is the difference of the Levi-Civita connection∇ of the metricg and the Levi-Civita connection ∇ of the metric g [8]. Notice that in [8] the author considered intrinsic torsion differing by the sign form the intrinsic torsion considered in this article and by other authors.…”

“…All the information in this section can be found in [3,8]. Let (M, g) be an oriented Riemannian manifold.…”

“…Let us state results obtained in [3,8] concerning harmonicity and minimality of G-structures. For any endomorphism T :…”

On the frame bundle adapted to a submanifold

“… Remark It seems that the correlation of minimality of and the harmonicity of associated section of the Grassmann bundle is a more general fact, which holds for any restriction of the structure group of the principal bundle and the homogeneous bundle associated with this bundle. This correlation is studied by the author independently .…”

“…In that case the adapted frame bundle is a bundle over N and the connection is a connection on N . These results and its generalizations are studied by the author independently .…”