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arXiv · 1909.09717

Introduction to $\mathrm{G}_2$ geometry

Spiro Karigiannis
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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.

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Spiro Karigiannis. 2019-09-20. Introduction to $\mathrm{G}_2$ geometry. https://doi.org/10.1007/978-1-0716-0577-6_1

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