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G2 Manifolds Seminar: Spiro Karigiannis

Title:Fundamentals of exceptional holonomy, I
Speaker:  Spiro Karigiannis
Date: Tuesady, August 19, 2014
Time: 09:30am-10:30am
Place: Seminar Room 313, Simons Center



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Abstract: This is part one of an introduction to the geometry of $\mathrm{G}_2$ and $\mathrm{Spin}(7)$ structures, henceforth called “exceptional structures”. We will begin with a very brief review of Berger’s list of Riemannian holonomy groups and of some non-exceptional structures on manifolds such as almost Hermitian structures. Then we will introduce the octonions, cross products, and the exceptional calibrations on $\mathbb R^7$ and $\mathbb R^8$, which will allow us to define exceptional structures on manifolds via the structure group of their frame bundles. Next, we will study the concrete representation theory of $\mathrm{G}_2$ and $\mathrm{Spin}(7)$, which will allow us to define the torsion forms and define various classes of exceptional structures. Finally, we will express the Ricci tensor in terms of the torsion, and give a concrete computational proof of the theorems of Fern\`andez-Gray and Fern\`andez relating parallel and harmonic calibration forms.