There is a rich theory of so-called (strict) nearly K¨ahler manifolds,
almost-Hermitian manifolds generalising the famous almost complex structure
on the 6-sphere induced by octonionic multiplication. Nearly K¨ahler
6-manifolds play a distinguished role both in the general structure theory
and also because of their connection with singular spaces with holonomy
group the compact exceptional Lie group G2: the metric cone over a Riemannian
6-manifold M has holonomy contained in G2 if and only if M is
a nearly K¨ahler 6-manifold.
A central problem in the field has been the absence of any complete
inhomogeneous examples. We prove the existence of the first complete
inhomogeneous nearly K¨ahler 6-manifolds by proving the existence of at
least one cohomogeneity one nearly K¨ahler structure on the 6-sphere and
on the product of a pair of 3-spheres. We conjecture that these are the
only simply connected (inhomogeneous) cohomogeneity one nearly K¨ahler
structures in six dimensions.