New $\mathrm{G}_2$-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 structureon the 6-sphere induced by octonionic multiplication. Namely 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 thecompact exceptional Lie group $\mathrm{G}_2$: the metric cone over aRiemannian 6-manifold M has holonomy contained in $\mathrm{G}_2$ if and onlyif $M$ 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 existenceof at least one cohomogeneity one nearly Kähler structure on the6-sphere and on the product of a pair of 3-spheres. We conjecture thatthese are the only simply connected (inhomogeneous) cohomogeneity one nearly Kähler structures in six dimensions.