### Abstract

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 G

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.

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

Original language | English |
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Pages (from-to) | 59-130 |

Number of pages | 71 |

Journal | Annals of Mathematics |

Volume | 185 |

Issue number | 1 |

Early online date | 2 Dec 2016 |

DOIs | |

Publication status | Published - 31 Jan 2017 |

### Keywords

- Einstein manifolds
- Exceptional holonomy
- Nearly Kahler 6 manifolds
- G_2-holonomy cone

### ASJC Scopus subject areas

- Mathematics(all)
- Geometry and Topology

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## Cite this

Foscolo, L., & Haskins, M. (2017). New G

^{2}-holonomy cones and exotic nearly Kähler structures on S^{6}and S^{3}×S^{3}.*Annals of Mathematics*,*185*(1), 59-130. https://doi.org/10.4007/annals.2017.185.1.2