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


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.


Lorenzo Foscolo

Stony Brook University, Stony Brook, NY

Mark Haskins

Imperial College London, South Kensington Campus, London, UK