Introduction to $\mathrm{G}_2$ geometry

TL;DR

This paper provides an accessible introduction to $ ext{G}_2$ geometry, focusing on its algebraic structure, geometric properties, and key theorems related to torsion-free $ ext{G}_2$ manifolds, suitable for beginners.

Contribution

It offers a comprehensive, beginner-friendly overview of $ ext{G}_2$ geometry, emphasizing algebraic structures, geometric insights, and important theorems, with connections to holonomy and special manifolds.

Findings

Explains the algebraic structure of $ ext{G}_2$ in 7 dimensions using octonions.

Discusses the relation between torsion, curvature, and holonomy in $ ext{G}_2$-structures.

Surveys key theorems about compact torsion-free $ ext{G}_2$ manifolds.

Abstract

These notes give an informal and leisurely introduction to $\mathrm{G}_2$ geometry for beginners. A special emphasis is placed on understanding the special linear algebraic structure in $7$ dimensions that is the pointwise model for $\mathrm{G}_2$ geometry, using the octonions. The basics of $\mathrm{G}_2$-structures are introduced, from a Riemannian geometric point of view, including a discussion of the torsion and its relation to curvature for a general $\mathrm{G}_2$-structure, as well as the connection to Riemannian holonomy. The history and properties of torsion-free $\mathrm{G}_2$ manifolds are considered, and we stress the similarities and differences with Kahler and Calabi-Yau manifolds. The notes end with a brief survey of three important theorems about compact torsion-free $\mathrm{G}_2$ manifolds.