Integrable G₂ Structures on 7-dimensional 3-Sasakian Manifolds

Yıl 2017, Cilt: 21 Sayı: 1, 254 - 260, 18.02.2017

Nülifer Özdemir , Şirin Akay

Öz

It is known that there exist canonical and nearly parallel $G_2$ structures on 7-dimensional 3-Sasakian manifolds. In this paper, we investigate the existence of $G_2$ structures which are neither canonical nor nearly parallel. We obtain eight new $G_2$ structures on 7-dimensional 3-Sasakian manifolds which are of general type according to the classification of $G_2$ structures by Fernandez and Gray. Then by deforming the metric determined by the $G_2$ structure, we give integrable $G_2$ structures. On a manifold with integrable $G_2$ structure, there exists a uniquely determined metric covariant derivative with anti-symetric torsion. We write torsion tensors corresponding to metric covariant derivatives with skew-symmetric torsion. In addition, we investigate some properties of torsion tensors.

Anahtar Kelimeler

G2 group, G2 structure; 3-Sasakian structure

Kaynakça

• [1] Bryant, R. L., Metrics with Exceptional Holonomy, Ann. of Math., 126 (1987), 525-586.
• [2] Fernández, M. and Gray, A., Riemannian Manifolds with Structure Group G2, Ann. Mat. Pura Appl. (4) 132 (1982) 19-25.
• [3] Cabrera, F. M., On Riemannian Manifolds with G2- Structure, Bolletino U.M.I (7) 10-A (1996) 99-112.
• [4] Boyer, C.P and Galicki, K., 3-Sasakian Manifolds, Surveys Diff. Geom., 7, (1999) 123-184, arXiv:hep-th/ 9810250.
• [5] Agricola, I. and Friedrich, T., 3-Sasakian Manifolds in Dimension Seven, Their Spinors and G2 Structures, Journal of Geometry and Physics 60 (2010) 326-332.
• [6] Friedrich,T.,Kath,I.,Moroianu,A.andSemmelmann, U., On Nearly Parallel G2 Structures, J. Geom. Phys., 23 (1997) 256-286.
• [7] Friedrich, T. and Ivanov, S., Parallel Spinors and Con- nections with Skew-Symmetric Torsion in String The- ory, Asian Jour. Math., 6 (2002) 303-336.
• [8] Agricola, I., The Srni Lectures on Non-Integrable Ge- ometries with Torsion, Math. Ann., 328 (2004) 711- 748.