$G_2$-manifolds and the ADM formalism

TL;DR

This paper explores the relationship between $G_2$-manifolds and the ADM formalism by analyzing a Hamiltonian system on the space of Riemannian metrics, revealing a correspondence with certain foliated $G_2$-manifolds.

Contribution

It establishes a novel connection between Hamiltonian dynamics on the space of metrics and the geometry of $G_2$-manifolds foliated by specific hypersurfaces.

Findings

Orbits of the constrained Hamiltonian system correspond to $G_2$-manifolds.

Foliations are diffeomorphic to $M \times \mathrm{SO}(3)$.

Provides a Hamiltonian perspective on $G_2$-geometry.

Abstract

In this paper we study a Hamiltonian function on the cotangent bundle of the space of Riemannian metrics on a 3-manifold $M$ and prove the orbits of the constrained Hamiltonian dynamical system correspond to $G_2$-manifolds foliated by hypersurfaces diffeomorphic to $M\times \mathrm{SO}(3)$.