# Metrics with exceptional holonomy

### Abstract

It is proved that there exist metrics with holonomy $G_2$ and $\mathrm{Spin}(7)$, thus settling the remaining cases in Berger’s list of possible holonomy groups. We first reformulate the “holonomy $H$” condition as a set of differential equations for an associated $H$-structure on a given manifold. We collect the needed algebraic acts about $G_2$ and $\mathrm{Spin}(7)$. We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kaähler theorem) to prove the existence of solutions whose holonomy is $G_2$ or $\mathrm{Spin}(7)$. We also provide explicit examples and some information about the “generality” of the space of such metrics.

Robert L. Bryant