The Space of Closed $G_2$-Structures. I. Connections

TL;DR

This paper develops the foundational geometric theory of the space of closed $G_2$-structures, including metrics, connections, geodesics, and variational analysis, to understand their infinite-dimensional manifold structure.

Contribution

It introduces Sobolev-type metrics and constructs Levi-Civita connections on the space of closed $G_2$-structures, advancing the mathematical understanding of their geometric properties.

Findings

Defined Sobolev-type metrics on the space of closed $G_2$-structures

Constructed Levi-Civita connections for these metrics

Analyzed geodesic equations and variational structures

Abstract

In this article, we develop foundational theory for geometries of the space of closed $G_2$-structures in a given cohomology class as an infinite-dimensional manifold. We introduce Sobolev-type metrics, construct their Levi-Civita connections, formulate geodesic equations, and analyse the variational structures of torsion free $G_2$-structures under these Sobolev-type metrics.