On geometry of the (generalized) G2-manifolds

Abstract

In this paper, we first understand the classical [Formula: see text]-structure and [Formula: see text]-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 [Formula: see text]-manifold, more precisely, we prove some no-go theorems corresponding to the vector field preserving the associated 4-form on [Formula: see text]-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 [Formula: see text]-structures. In the last section, we deal with the structure operators on generalized [Formula: see text]-manifold after describing the integrability of generalized [Formula: see text]-structure, some identities of structure operators are derived, which are analogues of Kähler-type and Weitzenböck-type identities under the classical case. And finally, we introduce a flow of which a generalized [Formula: see text]-manifold can be realized as the fixed point.