Introduction to G2 Geometry

Abstract

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