New G<inf>2</inf>-holonomy cones and exotic nearly Kahler structures on S<sup>6</sup> and S<sup>3</sup> x S<sup>3</sup>

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© 2017 Department of Mathematics, Princeton University. There is a rich theory of so-called (strict) nearly Kahler manifolds, almost-Hermitian manifolds generalising the famous almost complex structure on the 6-sphere induced by octonionic multiplication. Nearly Kahler 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 Kahler 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 Kahler 6-manifolds by proving the existence of at least one cohomogeneity one nearly Kahler 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 Kahler structures in six dimensions.





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Foscolo, L, and M Haskins (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 Retrieved from

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