Compact Riemannian manifolds with exceptional holonomy

Abstract

Suppose that M is an orientable n-dimensional manifold, and g a Riemannian metric on M . Then the holonomy group Hol(g) of g is an important invariant of g. It is a subgroup of SO(n). For generic metrics g on M the holonomy group Hol(g) is SO(n), but for some special g the holonomy group may be a proper Lie subgroup of SO(n). When this happens the metric g is compatible with some extra geometric structure on M , such as a complex structure. The possibilities for Hol(g) were classified in 1955 by Berger. Under conditions on M and g given in §1, Berger found that Hol(g) must be one of SO(n), U(m), SU(m), Sp(m), Sp(m)Sp(1), G2 or Spin(7). His methods showed that Hol(g) is intimately related to the Riemann curvature R of g. One consequence of this is that metrics with holonomy Sp(m)Sp(1) for m > 1 are automatically Einstein, and metrics with holonomy SU(m), Sp(m), G2 or Spin(7) are Ricci-flat. Now, people have found many different ways of producing examples of metrics with these holonomy groups, by exploiting the extra geometric structure – for example, quotient constructions, twistor geometry, homogeneous and cohomogeneity one examples, and analytic approaches such as Yau’s solution of the Calabi conjecture. Naturally, these methods yield examples of Einstein and Ricci-flat manifolds. In fact, metrics with special holonomy groups provide the only examples of compact, Ricci-flat Riemannian manifolds that are known (or known to the author). The holonomy groups G2 and Spin(7) are known as the exceptional holonomy groups, since they are the exceptional cases in Berger’s classification. Here G2 is a holonomy group in dimension 7, and Spin(7) is a holonomy group in dimension 8. Thus, metrics with holonomy G2 and Spin(7) are examples of Ricci-flat metrics on 7and 8-manifolds. The exceptional holonomy groups are the most mysterious of the groups on Berger’s list, and have taken longest to reveal their secrets – it was not even known until 1985 that metrics with these holonomy groups existed. The purpose of this chapter is to describe the construction of compact Riemannian manifolds with holonomy G2 and Spin(7). These constructions were found in 1994-5 by the present author, and appear in [16], [17] for the case of G2, and in