Compact 8-manifolds with holonomySpin(7)

Abstract

In Berger’s classification [4] of the possible holonomy groups of a nonsymmetric, irreducible riemannian manifold, there are two special cases, the exceptional holonomy groups G2 in 7 dimensions and Spin(7) in 8 dimensions. Bryant [6] proved that such metrics exist locally, using the theory of exterior differential systems, and gave some explicit examples. Later, Bryant and Salamon [7] constructed explicit, complete metrics with holonomy G2 and Spin(7). In two previous papers [12], [13], the author constructed many examples of compact riemannian 7-manifolds with holonomy G2. This paper will construct examples of compact riemannian 8-manifolds with holonomy Spin(7), using similar methods. We believe that these are the first examples known. Since metrics with holonomy Spin(7) are ricci-flat, these are also new examples of compact, ricci-flat riemannian 8-manifolds. Let M be an 8-manifold. A Spin(7)structure on M can be encoded in a 4-form Ω on M , a special 4-form satisfying the condition that the stabilizer of Ω at each point should be isomorphic to Spin(7). By an abuse of notation, we usually identify the Spin(7)structure with its associated 4-form Ω. Since Spin(7) ⊂ SO(8), the Spin(7)structure also induces a riemannian metric g and an orientation on M . It turns out that the holonomy group Hol(g) of g is a subgroup of Spin(7), with Spin(7)structure Ω, if and only if dΩ = 0. The quantity dΩ is called the torsion of the Spin(7)structure Ω, and Ω is called torsion-free if dΩ = 0. Very briefly, the plan of the paper splits into four steps, as follows. Firstly, a compact 8-manifold M is given. Secondly, a Spin(7)structure Ω on M is found, with small torsion, i.e. dΩ is small. Thirdly, Ω is deformed to a nearby Spin(7)structure Ω with dΩ = 0. Thus Ω is torsion-free, and if g is the associated metric then Hol(g) ⊂ Spin(7). Fourthly, it is shown that Hol(g) is Spin(7), and not some proper subgroup. The structure of the paper is designed around this division of the construction into four steps. There are six chapters. This first chapter is of introductory material. The second chapter has the full statements of the main results. In