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Introduction to \(\mathrm {G}_2\) Geometry

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Lectures and Surveys on G2-Manifolds and Related Topics

Part of the book series: Fields Institute Communications ((FIC,volume 84))

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 Kähler 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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Acknowledgements

The author would like to acknowledge Jason Lotay and Naichung Conan Leung for useful discussions on the structuring of these lecture notes. The initial preparation of these notes was done while the author held a Fields Research Fellowship at the Fields Institute. The final preparation of these notes was done while the author was a visiting scholar at the Center of Mathematical Sciences and Applications at Harvard University. The author thanks both the Fields Institute and the CMSA for their hospitality.

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  1. Department of Pure Mathematics, University of Waterloo, Waterloo, Canada

    Spiro Karigiannis

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  1. Spiro Karigiannis
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Correspondence to Spiro Karigiannis .

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Editors and Affiliations

  1. Department of Pure Mathematics, University of Waterloo, Waterloo, ON, Canada

    Spiro Karigiannis

  2. Institute of Mathematical Sciences, Chinese University of Hong Kong, Shatin, New Territories, Hong Kong

    Naichung Conan Leung

  3. Mathematical Institute, University of Oxford, Oxford, UK

    Jason D. Lotay

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Karigiannis, S. (2020). Introduction to \(\mathrm {G}_2\) Geometry. In: Karigiannis, S., Leung, N., Lotay, J. (eds) Lectures and Surveys on G2-Manifolds and Related Topics. Fields Institute Communications, vol 84. Springer, New York, NY. https://doi.org/10.1007/978-1-0716-0577-6_1

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