Metrics with exceptional holonomy. (English) Zbl 0637.53042

Bryant, Robert L.

Ann. Math. (2) 126, 525-576 (1987).

From the author’s abstract “It is proved that there exist metrics with holonomy \(G_ 2\) and 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 facts about \(G_ 2\) and Spin(7). We then apply the machinery of over-determined partial differential equations (in the form of the Cartan-Kähler theorem) to prove the existence of solutions whose holonomy is \(G_ 2\) or Spin(7). We also provide explicit examples and some information about the “generality” of the space of such metrics.”
Moreover, the explicit examples given in section 5, are cones on homogeneous spaces. For example, to settle the \(G_ 2\) case, the author shows that the normal SU(3)-invariant metric on \(SU(3)/T^ 2\), \(T^ 2\) a maximal torus, gives rise to a cone metric on \(R^+\times (SU(3)/T^ 2)\) with holonomy \(G_ 2\). An analogous construction is used in the Sp(7) case.

Reviewer: I.Dotti-Miatello

MSC:

53C10	\(G\)-structures
53C20	Global Riemannian geometry, including pinching

Keywords:

holonomy groups; over-determined partial differential equations; Cartan- Kähler theorem; homogeneous spaces