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

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.

v2: Minor correction to proof of inhomogeneity of new nearly Kaehler structure in Theorem 7.12. Added Remark 7.13 on further consequences of the revised argument. Added two further references. v3: Corrected several typos and minor imprecisions; made minor expositional improvements suggested by referee; streamlined Section 9. To appear in the Annals of Mathematics

References in corpus (1)

• Cohomogeneity one Einstein-Sasaki 5-manifolds

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