New G2-holonomy cones and exotic nearly Kähler structures on S6 and S3 x S3

Lorenzo Foscolo, Mark Haskins

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75 Citations (Scopus)
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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 G2: the metric cone over a Riemannian 6-manifold M has holonomy contained in G2 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.
Original languageEnglish
Pages (from-to)59-130
JournalAnnals of Mathematics
Issue number1
Publication statusPublished - 2 Dec 2016


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