Narrow Results

This paper is a review of current developments in the study of moduli spaces of G₂ manifolds. G₂ manifolds are seven-dimensional manifolds with the exceptional holonomy group G₂. Although they are odd-dimensional, in many ways they can be considered as an analogue of Calabi–Yau manifolds in seven dimensions. They play an important role in physics as natural candidates for supersymmetric vacuum solutions of M-theory compactifications. Despite the physical motivation, many of the results are of purely mathematical interest. Here we cover the basics of G₂ manifolds, local deformation theory of G₂ structures and the local geometry of the moduli spaces of G₂ structures.

In this paper, we show that a parallel differential form $\mathrm{\Psi }$ of even degree on a Riemannian manifold allows to define a natural differential both on ${\mathrm{\Omega }}^{\ast }(M)$ and ${\mathrm{\Omega }}^{\ast }(M,TM)$, defined via the Frölicher–Nijenhuis bracket. For instance, on a Kähler manifold, these operators are the complex differential and the Dolbeault differential, respectively. We investigate this construction when taking the differential with respect to the canonical parallel $4$-form on a $G_2$- and $\mathrm{\text{Spin}}(7)$-manifold, respectively. We calculate the cohomology groups of ${\mathrm{\Omega }}^{\ast }(M)$ and give a partial description of the cohomology of ${\mathrm{\Omega }}^{\ast }(M,TM)$.

We study the geometry of hypersurfaces in manifolds with Ricci-flat holonomy group, on which we introduce a G-structure whose intrinsic torsion can be identified with the second fundamental form. The general problem of extending a manifold with such a G-structure so as to invert this construction is open, but results exist in particular cases, which we review. We list the five-dimensional nilmanifolds carrying invariant SU(2)-structures of this type, and present an example of an associated metric with holonomy SU(3).

This article is a broad-brush survey of two areas in differential geometry. While these two areas are not usually put side-by-side in this way, there are several reasons for discussing them together. First, they both fit into a very general pattern, where one asks about the existence of various differential-geometric structures on a manifold. In one case we consider a complex Kähler manifold and seek a distinguished metric, for example a Kählera–Einsten metric. In the other we seek a metric of exceptional holonomy on a manifold of dimension 7 or 8. Second, as we shall see in more detail below, there are numerous points of contact between these areas at a technical level. Third, there is a pleasant contrast between the state of development in the fields. These questions in Kähler geometry have been studied for more than half a century: there is a huge literature with many deep and far-ranging results. By contrast, the theory of manifolds of exceptional holonomy is a wide-open field: very little is known in the way of general results and the developments so far have focused on examples. In many cases these examples depend on advances in Kähler geometry.