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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
Abstract:
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.
Keywords:
$G$-structure; intrinsic torsion; minimal submanifold; harmonic mapping.
MSC: 53C10;
53C24;
53C43;
53C15 Received: April 28, 2016; in final form
October 31, 2016; Published online
November 4, 2016
Language: English
DOI:
10.3842/SIGMA.2016.107
Fulltext:
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