New G$_2$-holonomy cones and exotic nearly Kähler structures on $S^6$ and $S^3 \times S^3$

Abstract

There is a rich theory of so-called (strict) nearly Kähler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the $6$-sphere induced by octonionic multiplication. Nearly Kähler $6$-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 $\mathrm{G}_2$: the metric cone over a Riemannian $6$-manifold $M$ has holonomy contained in $\mathrm{G}_2$ if and only if $M$ is a nearly Kähler $6$-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 Kähler $6$-manifolds by proving the existence of at least one cohomogeneity one nearly Kähler 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 Kähler structures in six dimensions.