New G2 holonomy cones and exotic nearly Kaehler structures on the 6-sphere and the product of a pair of 3-spheres

TL;DR

This paper constructs the first known complete inhomogeneous nearly Kaehler 6-manifolds, specifically on the 6-sphere and the product of two 3-spheres, advancing understanding of special geometric structures.

Contribution

It proves the existence of cohomogeneity one nearly Kaehler structures on these manifolds, providing new examples and addressing a longstanding gap in the field.

Findings

Existence of cohomogeneity one nearly Kaehler structures on the 6-sphere

Existence of such structures on the product of two 3-spheres

Conjecture that these are the only inhomogeneous examples in six dimensions

Abstract

There is a rich theory of so-called (strict) nearly Kaehler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kaehler 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 Kaehler 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 Kaehler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kaehler 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 Kaehler structures in six dimensions.