with Salamon, S; Bryant, RL, *On the construction of some complete metrics with exceptional holonomy*,
Duke Math. J., vol. 58 no. 3
(1989),
pp. 829-850 [MR90i:53055]

(last updated on 2018/10/14)
**Author's Comments:**

We construct examples of complete metrics with
holonomy G_{2}
and Spin(7). Specifically, on the product of the 3-sphere
with real 4-space,
we construct a complete SO(4)-invariant metric with
holonomy G_{2},
on the bundle of self-dual 2-forms on the complex
projective plane, we
construct a complete SU(3)-invariant metric with
holonomy G_{2},
and on the positive spin bundle over the 4-sphere, we
construct a complete
Spin(5)-invariant metric with holonomy Spin(7).

The method is to look among the group invariant
metrics for one that
has the right holonomy. The point, in each case, is that
the group acts
with cohomogeneity one and so the problem is
reduced to an ODE problem.
These ODE turn out to be managable. In many ways,
the construction is reminiscent
of Calabi's construction of a complete metric with
holonomy Sp(n) on the
holomorphic cotangent bundle of complex projective
n-space.

It is now known, from Dominic Joyce's work (see MR
99j:53065 and the references contained therein),
that compact examples
exist in both cases.

Recently, physicists S. Gukov and coworkers and
C. Page and coworkers (not to mention Atiyah and
Witten) have produced more examples of
cohomogeneity one metrics with exceptional holonomy.
Of course, in each case, it `reduces' to a study of a
certain ODE system, which can sometimes be
integrated.