Foscolo, L; Haskins, M; (2017) New G₂-holonomy cones and exotic nearly Kahler structures on S⁶ and S³ x S³. Annals of Mathematics , 185 (1) pp. 59-130. 10.4007/annals.2017.185.1.2.

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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 G_{2}: the metric cone over a Riemannian 6-manifold M has holonomy contained in G_{2} 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.

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