On seven dimensional 3-Sasakian manifolds

Abstract

3-Sasakian manifolds in dimension seven have cocalibrated andnearly parallel G-structures. In this work, cocalibrated G-structure is deformed by one of the characteristic vector fields of the 3-Sasakian structure anda new G2structure is obtained whose metric has negative scalar curvature. Inaddition, the new G2structure has a nonzero Killing vector field. Then, by using this deformation, new covariant derivative on the spinor bundle is obtained and the new Dirac operator is written in terms of the Dirac operator before deformation

Keywords

• 3-Sasakian manifold, Riemannian manifold with structure group G2

References

1. Fernández, M. and Gray, A., Riemannian manifolds with structure group G, Ann. Mat. Pura Appl., 4 (132), 19-25, 1982.
2. Karigiannis, S., Deformations of Gand Spin(7) Structures on Manifolds, Canadian journal of mathematics, 5, 1012-1055, 2005.
3. Boyer, C. and Galicki, K., Sasakian Geometry, Oxford University Press, 2008.
4. Boyer, C. P and Galicki, K., 3-Sasakian Manifolds, Surveys Diğ . Geom. 7, 123-184, 1999.
5. Friedrich, T. and Kath I., Varietes Riemanniennes compactes de dimension 7 admettant des spineurs de Killing, C. R. Acad. Sci Paris 307 Serie I, 967-969, 1988.
6. Friedrich, T. and Kath I., Compact seven dimensional manifolds with Killing spinors, Comm. Math. Phys., 133, 543-561, 1990.
7. Friedrich, T. and Kath I., Moroianu, A. and Semmelman, U., On nearly parallel G2structures, J. Geom. Phys., 23, 256-286, 1997.
8. Agricola, I. and Friedrich, T., 3-Sasakian manifolds in dimension seven, their spinors and G2 structures, J. Geom. Phys., 60, 326-332, 2010.

9. Friedrich, T., Dirac Operators in Riemannian Geometry, American Mathematical Society, Providence, 2000.
10. Hijazi, O., Spectral Properties of the Dirac operator and geometrical structures, Proceedings of the Summer School on Geometric Methods in Quantum Field Theory, Villa de Leyva, Colombia, July 12-30, 1999, World Scienti…c 2001.
11. Lawson H. B. and Michelsohn M. L., Spin Geometry, Princeton University Press, Princeton, New Jersey, 1989.
12. Özdemir N. and Aktay, S. On Deformations of G2Structures by Killing Vector Fields, Adv. Geom., 4 (14), 683-690, 2014.
13. Özdemir, N. and Aktay, S., Examples of Deformations of Nearly Parallel G2Structures on dimensional 3-Sasakian Manifolds by Characteristic Vector Fields, Int. Electron. J. Geom., , 31-37, 2014.
14. Current address : Department of Mathematics, Anadolu University, 26470 Eski¸sehir, Turkey E-mail address : [e-mail]