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

SIGMA 12 (2016), 107, 14 pages      arXiv:1503.03740      https://doi.org/10.3842/SIGMA.2016.107

### 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

Abstract
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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References

1. Besse A.L., Einstein manifolds, Classics in Mathematics, Springer-Verlag, Berlin, 2008.
2. Chinea D., Gonzalez C., A classification of almost contact metric manifolds, Ann. Mat. Pura Appl. (4) 156 (1990), 15-36.
3. Chiossi S., Salamon S., The intrinsic torsion of $\rm SU(3)$ and ${\rm G}_2$ structures, in Differential Geometry (Valencia, 2001), World Sci. Publ., River Edge, NJ, 2002, 115-133, math.DG/0202282.
4. Cleyton R., Swann A., Einstein metrics via intrinsic or parallel torsion, Math. Z. 247 (2004), 513-528, math.DG/0211446.
5. Davidov J., Muškarov O., Harmonic almost-complex structures on twistor spaces, Israel J. Math. 131 (2002), 319-332.
6. Eells Jr. J., Sampson J.H., Harmonic mappings of Riemannian manifolds, Amer. J. Math. 86 (1964), 109-160.
7. Falcitelli M., Farinola A., Salamon S., Almost-Hermitian geometry, Differential Geom. Appl. 4 (1994), 259-282.
8. Fernández M., Gray A., Riemannian manifolds with structure group ${\rm G}_{2}$, Ann. Mat. Pura Appl. (4) 132 (1982), 19-45.
9. Gil-Medrano O., Relationship between volume and energy of vector fields, Differential Geom. Appl. 15 (2001), 137-152.
10. Gil-Medrano O., González-Dávila J.C., Vanhecke L., Harmonicity and minimality of oriented distributions, Israel J. Math. 143 (2004), 253-279.
11. González-Dávila J.C., Harmonicity and minimality of distributions on Riemannian manifolds via the intrinsic torsion, Rev. Mat. Iberoam. 30 (2014), 247-275.
12. González-Dávila J.C., Martín Cabrera F., Harmonic $G$-structures, Math. Proc. Cambridge Philos. Soc. 146 (2009), 435-459.
13. Gray A., Hervella L.M., The sixteen classes of almost Hermitian manifolds and their linear invariants, Ann. Mat. Pura Appl. (4) 123 (1980), 35-58.
14. Han D.-S., Lee E.-H., Harmonic Gauss map and Hopf fibrations, J. Korea Soc. Math. Educ. Ser. B Pure Appl. Math. 5 (1998), 55-63.
15. Martín Cabrera F., Special almost Hermitian geometry, J. Geom. Phys. 55 (2005), 450-470, math.DG/0409167.
16. Martín Cabrera F., Swann A., The intrinsic torsion of almost quaternion-Hermitian manifolds, Ann. Inst. Fourier (Grenoble) 58 (2008), 1455-1497, arXiv:0707.0939.
17. Naveira A.M., A classification of Riemannian almost-product manifolds, Rend. Mat. (7) 3 (1983), 577-592.
18. Niedziałomski K., On the frame bundle adapted to a submanifold, Math. Nachr. 288 (2015), 648-664, arXiv:1311.6172.
19. Vergara-Diaz E., Wood C.M., Harmonic almost contact structures, Geom. Dedicata 123 (2006), 131-151, math.DG/0602533.
20. Wood C.M., Harmonic almost-complex structures, Compositio Math. 99 (1995), 183-212.
21. Wood C.M., Harmonic sections of homogeneous fibre bundles, Differential Geom. Appl. 19 (2003), 193-210.