Nearly-Kähler $6$-manifolds of cohomogeneity two: principal locus

We study nearly-Kähler $6$-manifolds equipped with a cohomogeneity-two Lie group action for which the principal orbits are coisotropic. If the metric is complete, then we show that this last condition is automatically satisfied, and both the acting Lie group and the principal orbits are finite quotients of $\mathbb{S}^3 \times \mathbb{S}^1$.

We then partition the class of such nearly-Kähler structures into three types (called I, II, III) and prove a local existence and generality result for each type. Metrics of Types I and II are shown to be incomplete.

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Received 29 October 2018

Accepted 23 August 2019

Published 14 December 2022