# Introduction to $\mathrm{G}_2$~geometry, Part I

We begin with a very brief review of Berger's list of Riemannian holonomy groups and of the more well-known $\mathrm{U}(m)$-structures. Then we will introduce the octonions, cross products, and the exceptional calibrations on $\mathbb R^7$, which will allow us to define $\mathrm{G}_2$-structures. Next, we will study the concrete representation theory of $\mathrm{G}_2$, which will allow us to define the torsion forms and define various classes of $\mathrm{G}_2$-structures. Finally, we will end part one by expressing the Ricci tensor in terms of the torsion, and give a concrete computational proof of the theorem of Fernandez-Gray relating parallel and harmonic calibration forms. In part two, we will briefly mention Joyce's perturbative existence theorem of torsion-free $\mathrm{G}_2$-structures given appropriate initial data used for compact constructions of smooth compact $\mathrm{G}_2$ manifolds. This topic will be treated in great detail in the later lectures of Kovalev. Finally, we will establish the smoothness of the moduli space of compact $\mathrm{G}_2$ manifolds and discuss some special geometric structures on this moduli space.