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

Lorenzo Foscolo, Mark Haskins

Research output: Contribution to journalArticle

35 Citations (Scopus)

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 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
Number of pages71
JournalAnnals of Mathematics
Volume185
Issue number1
Early online date2 Dec 2016
DOIs
Publication statusPublished - 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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