Remarks on Exact G$_{2}$-Structures on Compact Manifolds

TL;DR

This paper investigates whether compact seven-manifolds can support exact G$_{2}$-Structures, exploring their properties and relationships with other geometric conditions, addressing a key open question in G$_{2}$ geometry.

Contribution

It initiates a study on exact G$_{2}$-Structures on compact manifolds, analyzing their existence and relation to other G$_{2}$-structure conditions.

Findings

Examines the existence of exact G$_{2}$-Structures on compact manifolds

Analyzes the relationship between exact G$_{2}$-Structures and other conditions like Ricci-pinched and Laplacian solitons

Provides insights into the open question of support for exact G$_{2}$-Structures on compact seven-manifolds

Abstract

An important open question in G$_{2}$ geometry concerns whether or not a compact seven-manifold can support an exact G$_{2}$-Structure. Given the significance of this question we initiate a study of exact G$_{2}$-Structures on compact manifolds. We focus on exact G$_2$-Structures subject to no additional constraints but we also consider the relationship between the exact condition and other conditions for closed G$_{2}$-Structures such as the Extremally Ricci-Pinched and Laplacian Soliton conditions.