Some A3/7 deformations of G2 structures

In this thesis, a 7-dimensional Riemannian manifold with $G_2$ structure is considered. The fundamental 3-form of such a manifold is deformed by using an arbitrary vector field and the class change of the new $G_2$ structure is investigated. First, under some restrictions, the Levi-Civita covariant derivative, the spinorial covariant derivative and the Dirac operator on the spinor bundle are computed and it is proved that kernels of the old and the new Dirac operators are isomorphic. Next, the new Levi-Civita covariant derivative is expressed without any restriction. It is observed that if the vector field used while deforming the fundamental 3-form is a Killing vector field of unit length, then the formula derived for the Levi-Civita covariant derivative simplifies. Thus deformations on 7-dimensional 3-Sasakian manifolds are studied because of the existence of $G_2$ structures and Killing vector fields of unit length on them. In addition, the new $G_2$ structures are constructed on 7-dimensional 3-Sasakian manifolds and then some deformations of these $G_2$ structures are considered.