On geometry of the (generalized) G2-manifolds

In this paper, we first understand the classical G(2)-structure and G(2)-geometry from the viewpoint of spinor, which is a more familiar framework for physicists. Explicit construction of invariant spinor is given via the Dirac gamma matrices. We introduce a notion of multispinor bundle associated with invariant spinor and differential operator on the section of this bundle. Then we study the vector fields satisfy some additional properties on G(2)-manifold, more precisely, we prove some no-go theorems corresponding to the vector field preserving the associated 4-form on G(2)-manifold, and we also consider the nowhere-vanishing vector field which induces an integrable complex structure on the vertical direction of tangent bundle. Next we discuss the relation between the variation of metric and that of effective action on the moduli space of integrable G(2)-structures. In the last section, we deal with the structure operators on generalized G(2)-manifold after describing the integrability of generalized G(2)-structure, some identities of structure operators are derived, which are analogues of Kahler-type and Weitzenbock-type identities under the classical case. And finally, we introduce a flow of which a generalized G(2)-manifold can be realized as the fixed point.