Fundamentals of exceptional holonomy, I

Spiro Karigiannis

This is part one of an introduction to the geometry of G_2 and 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 R^7 and 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 G_2 and 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 Fernandez-Gray and Fernandez relating parallel and harmonic calibration forms.

http://scgp.stonybrook.edu/video_portal/video.php?id=1525

Simons- Program: G2 manifolds