Some observations on conformal symmetries of G 2-structures

Abstract

On a 7-manifold with a G 2-structure, we study conformal symmetries — which are vector fields whose flow generate conformal transformations of the G 2-structure. In particular, we focus on compact 7-manifolds and the condition that the Lee form of the G 2-structure is closed. Among other observations, we show that conformal symmetries are determined within a conformal class of the G 2-structure by the symmetries of a unique (up to homothety) G 2-structure whose Lee form is harmonic. On a related note, we also demonstrate that symmetries are split along fibrations when the Lee vector field is itself a symmetry.