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

• 成果类型: Article
• 署名作者: Foscolo, Lorenzo; Haskins, Mark
• 刊物名称: ANNALS OF MATHEMATICS
• ISSN/ISSBN: 0003-486X
• DOI: 10.4007/annals.2017.185.1.2
• 发表日期: 2017
• 页码: 59-130
• 关键词: cohomogeneity one; conical singularities; exceptional holonomy; einstein-metrics; killing spinors; RICCI CURVATURE; MANIFOLDS; compact; SPACES; 6-manifolds

摘要

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 G(2) : the metric cone over a Riemannian 6-manifold M has holonomy contained in G(2) if and only if M is a nearly Kahler 6-manifold. A central problem in the fi eld has been the absence of any complete inhomogeneous examples. We prove the existence of the fi rst 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.