Diffeomorphisms of 7-Manifolds with Closed G_2-Structure

Abstract

We introduce G 2 -vector fields, Rochesterian 1-forms and Rochesterian vector fields on manifolds with a closed G 2 -structure as analogues of symplectic vector fields, Hamiltonian functions and Hamiltonian vector fields respectively, and we show that the spaces X G 2 of G 2 -vector fields and X Roc of Rochesterian vector fields are Lie subalgebras of the Lie algebra of vector fields with the standard Lie bracket. We also define, in analogy with the Poisson bracket on smooth real-valued functions from symplectic geometry, a bracket operation on the space of Rochesterian 1-forms Ω 1Roc associated to the space of Rochesterian vector fields and prove, despite the lack of a Jacobi identity, a relationship between this bracket and diffeomorphisms which preserve G 2 -structures.