Diffeomorphisms of 7-Manifolds with Closed G_2-Structure

TL;DR

This paper introduces new geometric structures on 7-manifolds with closed G_2-structures, defining analogues of Hamiltonian vector fields and functions, and explores their algebraic properties and relation to diffeomorphisms.

Contribution

It defines G_2-vector fields, Rochesterian 1-forms, and Rochesterian vector fields, establishing their Lie algebra structures and a bracket operation analogous to the Poisson bracket.

Findings

G_2-vector fields form a Lie subalgebra

Rochesterian vector fields form a Lie subalgebra

A bracket operation on Rochesterian 1-forms relates to G_2-structure-preserving diffeomorphisms

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 of G_2-vector fields and 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 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.