Introduction to ${\mathrm{G}}_2$ geometry

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.

37 pages. To appear in a forthcoming volume of the Fields Institute Communications, entitled "Lectures and Surveys on G2 manifolds and related topics". Version 2: Corrected the references. No other changes

References in corpus (4)

• A new construction of compact torsion-free $G_2$-manifolds by gluing families of Eguchi-Hanson spaces
• Curvature decomposition of G_2 manifolds
• Hodge Theory for G2-manifolds: Intermediate Jacobians and Abel-Jacobi maps
• Bubble Tree Convergence of Conformally Cross Product Preserving Maps

Cited by in corpus (7)

• Observations on BPS observables in 6d
• $\mathcal{S}\mathcal{W}$-algebras and strings with torsion
• The heterotic G${}_2$ system with reducible characteristic holonomy
• Gradient estimates and parabolic frequency under the Laplacian G_2 flow
• A variational characterization of calibrated submanifolds
• Curvature pinching estimate under the Laplacian G_{2} flow
• Extrinsic geometry of calibrated submanifolds