Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 12 (2016), 107, 14 pages      arXiv:1503.03740

Geometry of $G$-Structures via the Intrinsic Torsion

Kamil Niedziałomski
Department of Mathematics and Computer Science, University of Łódź, ul. Banacha 22, 90-238 Łódź, Poland

Received April 28, 2016, in final form October 31, 2016; Published online November 04, 2016

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

Key words: $G$-structure; intrinsic torsion; minimal submanifold; harmonic mapping.

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